RADICAL CLASSES OF BOOLEAN ALGEBRAS by THEODORE ALEXANDER GALAY McAo, University of Manitoba s, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mathematics We accept this thesis as conforming to the reauired standard ^, i THE UNIVERSITY OF BRITISH COLUMBIA July, 1974 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of B r i t i s h Columbia Vancouver 8, Canada Date CCt^^t^LZ^ /S~ /f7? ASSTRACT This thesis obtains information about Boolean algebras means of the r a d i c a l concept. by One group of r e s u l t s revolves about the concept, theorems, and constructions of general r a d i c a l theory. We obtain some subdirect product representations by methods suggested by the theory. A large number of s p e c i f i c radicals are defined, and their properties and i n t e r - r e l a t i o n s h i p s are examined. This provides a natural frame-work for r e s u l t s describing what epimorphs an algebra can have. Some new r e s u l t s of t h i s nature are obtained intheprocess.Finally,acontributionismadetothes t r u c t u r e theory of complete Boolean algebras. Product decomposition theorems are obtained, some of which make use of chains of r a d i c a l classes. TABLE OF CONTENTS INTRO DUCT ION oe»*fle»Qecoee>eee«O0eo9oeoo*OQ&po»»«o6»«0e»*oe0«e»ee PRELIMINARIES CHAPTER ONE: ........ coco o » BO 9»e«ooeoce » » o • • e 0 o o e e o * o c © e« e • • • • RADICAL CLASSES OF BOOLEAN ALGEBRAS §1 » GenSX'Sl Rad X C a l Ill&Oir V §2. Radical Classes of Boolean Rings §3. Radical Classes of Boolean Algebras §4» Conventions and Summary CHAPTER TWO: I oe.cti.»«eooo«oo.e.0»e«.vo.o«o.e „..... ...... . , . . . . . .••.«»«•••••.>.«.o \& 21 23 OIMIKMI 26 .....o..o*.o........... 28 THE LOWER RADICAL 51. The Lower Radical Construction §2. The Lower Radical f o r Boolean Algebra Radicals §3 c T ll£ SilpGTTfitlOIIiiC R&d1C&1 §4 e The C a r d i n a l i t y Radicals 55 o T llS POWGJT^Sst Radical S CHAPTER THREE: 30 . 0o?««eoo«eo«o«eo*eoe6 0 a « 6 o o e « « •••» o e e 33 « o o « * » D « e » « e o « e » « * 38 o e 0 e oof>OQ«o«>6ceoo»?**o*«ee«O0*eoe*'O 40 THE UPPER RADICAL §1. The Upper Radical Construction ........<>.............. 44 52. The Characterization Theorem .....r 45 §3. Atomless Boolean Algebras §4. Some Upper Radicals §5. The Upper Radical Determined by Homogeneous Algebras ............................ 46 .................................. 47 50 CHAPTER FOUR: CRAMER'S RADICALS § 1 . . The Classes C 92. §3, -The The Radical V n Xl)2 0 0 0 0 0 . 0 0 0 0 . .09, 0 . 0 . . . . 0 . 0 0 0 0 000 0000 as Upper Radicals CHAPTER FIVE: §10 a coo........ 55 e o 00 ee te 00 o © o o ti o © •» 0 0 00 00 00 DECOMPOSITIONS OF COMPLETE ALGEBRAS GCHSITGLI SGttinf^ 00000000000000000 00000000000000 00 Decompositions into Homogeneous Algebras §3. Decompositions into Unequivocal Algebras §4, Connections with Cardinal Properties SI. 54 •»0v6e««*oo«o«eooe««««««e9*e« 52, CHAPTER SIX: 53 «• o o o 37 59 . 61 •....•.»«••.. 63 «•.....•....•..•. CLOSURE PROPERTIES OF RADICAL AND SEMI-SIMPLE CLASSES Closure of Radical Classes under Products 67 v..».«•«.... J.2. .Closure,of.Radical j.Clas.see,,undsr ,C,QpXQdu,cts ...„.,............. 69 §3. Coproducts and Semi-Siroplicity CHAPTER SEVEN: THE LATTICE OF RADICALS §1. Lattice-Theoretic Preliminaries §2. The L a t t i c e of Radicals for Associative Rings §3. The L a t t i c e of Radicals f o r Boolean Algebras §4. Dual Atoms and Complements i n Lat(A) §5. Locating Known Radicals i n Lat(A) §6. Atoms i n the L a t t i c e §7. A Diagram of the L a t t i c e 0, 79 .00....0000.0.0 83 00000 . .........o...0000.000000. o o •«0 83 88 • 0000 89 93 00000 .....> Q C 0 « • « O > « M OOOO 0 00000 BIBLIOGRAPHY 75 000.00.00.000...00.000. 94 1 *~ e 0 * 0 0 0 * 0 • 00*00000 ee 00 0 * • « 0000*0 0*0000 « 0 e o o < • 0 0 0 0 INDEX OF NOTATION . . . . . . . . . . . . . . . . . . . . O 0 0 0O0OO0OO0O00 000 0 0 0 0 0 0 0 96 W 98 ACKNOWLEDGEMENTS I wish to express my appreciation to my thesis supervisor,, Dr. To Crauier;, for his continual encouragement and help during the preparation of t h i s t h e s i s . My sincere thanks also go to D r A, Adler f o r h i s f r i e n d l y support and constant c to enter into discussion, willingness I would al-so l i k e to thank the members of the Radical Theory Seminar, Dr. N, Divinsky, Dr 1, Anderson c and Dr, M, S l a t e r , 9 INTRODUCTION The f i r s t s i g n i f i c a n t breakthrough i n the study of s t r u c ture by means of r a d i c a l s was Cartan's c l a s s i f i c a t i o n of the f i n i t e dimensional semi-simple L i e algebras over the f i e l d of complex numbers. Early i n the twentieth century, Wedderburn obtained h i s struc- ture theorem f o r finite-dimensional associative algebras. nearly f o r t y years f o r the next major development: the Jacobson r a d i c a l and the density theorem. I t took the d e f i n i t i o n of The general procedure i n these cases was the same: f o r a c l a s s of rings„ define an i d e a l for each ring i n the c l a s s , c a l l i t the r a d i c a l said about r a d i c a l - f r e e (semi-simple) rings. p and see what can be In the early 1950's, Kurosh and Amitsur defined the general concept of a r a d i c a l c l a s s , which became the subject of much subsequent research. In 1939, Mostowski and Tarski introduced the notion of a superatoiaic Boolean algebra* A number of s i g n i f i c a n t r e s u l t s con- cerning t h i s class of algebras have been developed since, without e x p l o i t i n g the fact that i t i s a r a d i c a l c l a s s . In 1972, Cramer gener- a l i z e d the superatomic r a d i c a l to obtain a t r a n s f i n i t e chain of r a d i cal classes of Boolean algebras. Again, the ideas and methods were not r a d i c a l - t h e o r e t i c i n nature. The aim of this thesis i s to obtain information about Boolean algebras by means of the r a d i c a l concept, and to place these r e s u l t s in the setting of general r a d i c a l theory. In general r a d i c a l theory, i t Is necessary a c l a s s of rings closed under ideals and epimorphs. to begin with The c l a s s of Boolean algebras i s not such a c l a s s : an i d e a l of a Boolean algebra i s not a Boolean algebra. In Chapter Gne„ s t a r t i n g with the more general concept of a Boolean r i n g p we provide a natural d e f i n i t i o n of a r a d i c a l c l a s s of Boolean algebras. In the process, we obtain two i n t e r e s t i n g r e s u l t s conceniing these r a d i c a l s : f i r s t , every non-zero r a d i c a l class of Boolean algebras contains the c l a s s of superatouiic algebras; secondly, every r a d i c a l c l a s s of Boolean algebras i s hereditary. In Chapter Two, lower r a d i c a l . we investigate the construction of the In general,, t h i s i s an i n f i n i t e procedure; f o r Boolean algebras, the construction terminates at the second stage. If the generating c l a s s i s closed under epimorphs, we f i n d that a complete r a d i c a l algebra srust be a product of generating algebras. some lovrer r a d i c a l s , the most important radicals. We define of which we c a l l power-set For any power-set r a d i c a l , we obtain necessary and suffic- ient conditions for a complete algebra to be i n the r a d i c a l , and f o r an algebra to generate the r a d i c a l . The l a t t e r r e s u l t proves useful i n obtaining a product decomposition theorem f o r c e r t a i n complete algebras. In Chapter Three, we study the upper r a d i c a l construction. The algebras which are semi-simple with respect to an upper r a d i c a l are characterized as subdirect products of the algebras which determine the r a d i c a l . An upper r a d i c a l d e s c r i p t i o n of the superatoiaic r a d i c a l y i e l d s a c h a r a c t e r i z a t i o n of atomless Boolean algebras as subdirect products of separable, atomless algebras. Other upper radicals are defined, and we obtain a subdirect representation f o r the complete, atomless algebras. Chapter Four concerns the r a d i c a l defined by Cramer. Radical- theoretic methods are used to prove and extend some of h i s r e s u l t s . In p a r t i c u l a r , an upper r a d i c a l d e s c r i p t i o n i s obtained f o r some of h i s radicals. Pierce has conjectured that any complete Boolean algebra i s a product of homogeneous algebras. results i n t h i s d i r e c t i o n . Chapter Five gives some p a r t i a l I t i s shown that i f any descending chain of p r i n c i p a l i d e a l s i n a complete algebra has only f i n i t e l y many isomorphism types, then the algebra i s a product ras. of homogeneous algeb- The main result asserts that c e r t a i n complete algebras are products of unequivocal algebras (that i s , algebras which must be either r a d i c a l or semi-simple with respect to any r a d i c a l c l a s s ) . Some of these r e s u l t s are re-stated i n the language of c a r d i n a l prope r t i e s , which provides some a d d i t i o n a l i n s i g h t . Chapter Six gives some closure properties of r a d i c a l and semisimple classes under the formation of products and coproducts. Not unexpectedly, power-set r a d i c a l s are shown to be closed under suitably r e s t r i c t e d products of complete algebras. A new product i s defined which y i e l d s a r a d i c a l algebra whenever a l l the algebras involved i n the construction are r a d i c a l , A number of r a d i c a l classes are shown to be closed under f i n i t e coproducts. ing We obtain two r e s u l t s i n d i c a t - that coproducts are strongly related to semi-simplicity. First, any coproduct of algebras, one of which i s semi-simple, i s i t s e l f semi-simple. Secondly, f o r any r a d i c a l c l a s s , there i s a c a r d i n a l K such that f o r any algebra A of more than two elements, the coproduct of a t l e a s t K copies of A w i l l be semi-simple. In Chapter Seven, we regard r a d i c a l classes as elements of a lattice. The structure of this l a t t i c e i s investigated. .of Chapter*One i s r e - i t e r a t e d by.showln.the-isomorphism-of The theme the l a t t i c e of Boolean r i n g r a d i c a l s with the l a t t i c e of Boolean algebra r a d i c a l s . F i n a l l y , we focus on the s p e c i f i c r a d i c a l s defined i n t h i s work, with a view to l o c a t i n g them i n the l a t t i c e . PRELIMINARIES This section o u t l i n e s the terminology, notation, and basic f a c t s to be used. Basic references are Sikorski [24] and Halmos [13]. Further references are given as needed, and d e t a i l s are provided f o r results not e a s i l y accessible or e x p l i c i t l y stated i n the l i t e r a t u r e . §1 a) 0 Fundamental Notions We assume f a m i l i a r i t y with the concept of Boolean r i n g , both i n the ring-theoretic and l a t t i c e - t h e o r e t i c settings. Vie w i l l use both the r i n g operations (+, • ) , and the l a t t i c e operations of j o i n ( v ) , meet ( a ) , and complementation ('). % settings coincide. The concepts of ideal in the two We w i l l use the notation A x i d e a l of the r i n g A generated by the element x. f o r the p r i n c i p a l The term Boolean algebra (or...sinrply algebra) w i l l be used for a Boolean ring with a unity 1 d i s t i n c t from the zero 0. A subalgebra of a Boolean algebra i s a subring containing the u n i t y . Any ideal of a Boolean algebra gen- erates a subalgebra, consisting of the ideal together with the complements of elements of the i d e a l . ideal-generated subalgebra. Such a subalgebra w i l l be c a l l e d an An algebra-homomorphism i s a ring-homo- morphlsm which i s 1-preserving. b) The important connections between Boolean rings and algebras are the following: i ) every non-zero p r i n c i p a l i d e a l of a Boolean r i n g i s a Boolean algebra, i i ) any Boolean r i n g can be embedded as a maximal ideal i n a Boolean algebra, which i s unique up to isomorphism, and i i i ) any non-zero epimorph of a Boolean algebra i s i t s e l f an algebra, and the epimorphism must be 1-preserving. c) Vie w i l l use the symbol A f o r the class of a l l Boolean algebras, and the symbol B f o r the c l a s s of a l l Boolean rings, d) We assume f a m i l i a r i t y , also, with the Stone d u a l i t y theory, which assigns to any Boolean algebra A, a topological space S(A), c a l l e d i t s Stone space, which i s Bausdorff, compact, and t o t a l l y disconnected. The correspondence i s r e v e r s i b l e , and allows the following interchange of algebraic and topological concepts: i ) an element x of A (or the p r i n c i p a l i d e a l A ) x corresponds to a clopen subset S(x) of S(A), and A i s isomorphic to the algebra of clopen subsets of S(A); i i ) an ideal I of A corresponds to an open subset S(I) of S(A); namely 6 S(I) i s the union of the clopen sets S(x) f o r x e I; i i i ) the epimorph A/1 of A corresponds to the closed subset S(A) - S(I) of S(A); i v ) an embedding A >-> B corresponds to a continuous surjection S(B) — » S(A); v) an epimorphism A — » E corresponds to a continuous i n j e c - t i o n S(B)^-> S(A), e) L a t t i c e s w i l l always be assumed to have extreme elements 0 and 1„ which are d i s t i n c t . An atom i n a l a t t i c e i s a non-zero element which contains only 0 and i t s e l f . A dual atom i s an element d i s t i n c t from 1 which i s contained only i n 1 and i t s e l f . In the Stone space of a Boolean algebra, atoms appear as isolated points. a l a t t i c e are d i s j o i n t If x A y • 0, d i s t i n c t elements of i t are d i s j o i n t . Elements x and y of A set D i s d i s j o i n t e d i f any two The supreinutn of an a r b i t r a r y i n a l a t t i c e , when i t e x i s t s , w i l l be denoted by sup E. E The terms "•complete" (a-cotriplete)'- w i l l -b'fer u'se'd "to -indicate t h a f a r b iferary - ('count1 able) suprema always e x i s t . f) The Axiom of Choice w i l l be used without further mention, but the assumption of the Generalized Continuum Hypothesis (GCH) w i l l always be e x p l i c i t . For any cardinal K, we w i l l use the notation K+ f o r the next largest c a r d i n a l , and exp K f o r the cardinal more commonly denoted by 2 . K 52* Properties o£_ Boolean Rings and Algebras a) If I Is an ideal of a Boolean r i n g A s and J i s an ideal of I, then J i s an ideal of A. b) Every non-zero Boolean r i n g has a two-element denote the two-element c) epimorph. We w i l l Boolean algebra by 2. For any c o l l e c t i o n {A^ i e 1} of Boolean rings (algebras), the Cartesian product of the underlying sets together with the point-wise operations forms a Boolean r i n g (algebra), which we c a l l the product of the A^ and denote by A ed by x A2 s « " x A „ n 02 n(A^s i e I ) . F i n i t e products are denotThe Stone space of the product of algebras i s the Stone-Cech compactificatioa of the d i s j o i n t union of the S(A^), d) The subset of the product A of algebras A^, consisting of elements which are 0 i n a l l but a f i n i t e number of coordinates i s an i d e a l of A but i s not an algebra. The subalgebra of A which i t generates w i l l be called the weak product of the A^, and w i l l be denoted by the symbol wII(A^s 1 e I ) , I t s Stone space i s the one-point compactifi- cation of the d i s j o i n t union of the S(A^). e) An algebra A i s a r e t r a c t of an algebra B i f there i s an embedding f of A into B, and an epimorphism g of B onto A such that the composit i o n gf i s the i d e n t i t y on B. a retract of A, Any p r i n c i p a l i d e a l of an algebra A i s In p a r t i c u l a r , any factor algebra i n a product or weak product A i s a p r i n c i p a l i d e a l of A and so i s a r e t r a c t of A, f) Any Boolean r i n g A admits a product decomposition at any element x: namely, A i s the product of A from x. with the i d e a l of elements d i s j o i n t If A i s an algebra, we have A - k x A„«, x g) A subset D of a Boolean algebra A i s said to be dense i f i t con- s i s t s of non-zero elements, and any non-zero element of A contains an element of D, In p a r t i c u l a r , we can speak of dense ideals and dense subalgebras of an algebra. h) A Boolean algebra i s said to be separable i f i t lias a countable, dense subset. Any separable algebra has c a r d i n a l i t y at most exp H . Q Furthermore, there i s only one complete, atomless, separable algebra: the quotient of the algebra of Borel sets of r e a l numbers, modulo the ideal of a l l meagre Borel sets [16]. Proposition 1_: Let A be a subalgebra of a Boolean algebra B. Then B has an epimorph which has a dense subalgebra isomorphic to A. Proof; By Zorn's Lemma, choose an ideal I of B maximal with respect to the property A H I = 0. The r e s t r i c t i o n of the natural epimorphism f : B — » B/I to A i s an embedding, so B/I has a subalgebra isomorphic to A. Now suppose f ( b ) i s a non-zero element of B/I. Then b i s not i n I, so the ideal J generated by I and b i s s t r i c t l y larger than I . By the maximality of I, there i s a non-zero element a i n A fl J . Being i n J , t h i s element must have the form a = x v c, where x e I and c S b, Since f i s an embedding, 0 ^ f ( a ) f (c) ^ £ (b). m Hence the copy of A i n B/I i s denser S3 Comgljct^ Algebras 0 a) The algebra of a l l subsets of a set X w i l l be denoted by P(X)„ i s c a l l e d a power-set algebra. |x| and It i s characterized as the product of copies of the two-element algebra, and i t s Stone space i s the Stone- Cech compactiflcation gX of the set X with the discrete topology. When |x| = H b) Any product of algebras over an index set I has P(I) as a r e t r a c t , c) Complete algebras are p r e c i s e l y the r e t r a c t s of power-set algebras. a i s i n f i n i t e , we w i l l denote P (X) by P. a They'can also be characterized as 'the •Inject lye "Boolean algebras. An algebra C i s said to be i n j e c t i v e i f , A > g > B whenever f i s a horaomorphism from A to / £ / C C and g i s an embedding of A i n B, then there i s a homomorphism h of B to C such that f = hg, d) Pierce [22] has shown that an i n f i n i t e c a r d i n a l H i t y of a complete Boolean algebra i f and only i f H a e) i s the c a r d i n a l - a Any ° 'U » c 0i Boolean algebra A has a normal completion A with the following properties: i ) A i s complete, and contains A as a dense subalgebra. It i s unique with respect to these two properties, i i ) If A i s a subalgebra of a complete algebra B, then B has A as a subalgebra containing A, Using i ) , i t i s easy to see that i f x e A, then the normal completion of A i s the. p r i n c i p a l ideal of A generated by x. Hence x x ° f) An algebra A s a t i s f i e s the countable chain condition ( c , c . c ) i f any d i s j o i n t e d Proposition 2_s subset of A i s at most countable. If A i s an i n f i n i t e algebra s a t i s f y i n g c . c . c , then |A| - |A| °. H Proof: Every element of A can be represented as a d i s j o i n t (hence, at most countable) supremuro of elements from A. Also, g) |A| ° 4 |A| ° » Hence | A | < |A| ° , |A|, by d) above. Any o-complete algebra s a t i s f i e s the following condition: If the algebra A i s isomorphic to i t s p r i n c i p a l i d e a l A , then i t i s also isomorphic to A^ f o r any y £ x. h) A Boolean algebra A i s said to be homogeneous i f i t i s isomorphic to each of i t s non-zero p r i n c i p a l ideals. i) The following proposition i s suggested by Theorem 3.3 of Pierce {22]: Proposition 3: The normal completion of a homogeneous algebra i s homogeneous. Proof: A, Let A be homogeneous and suppose x i s a non-zero element of Choose some non-zero y i n A such that y < x. Since A^ - A, we get A^ - A by the uniqueness of the normal completion. Using g), we conclude that A j) Pierce (see Cramer [5]) a o-algebra has P a) - A. If Q has shown that any i n f i n i t e epimorph of as a r e t r a c t . i s the Stone space of A^ f o r i e I, then the topological product X of the X^ i s also a Boolean space. The algebra A of clopen supsets of X i s called the coproduct of the and w i l l be denoted by E(A^: i e I ) , For f i n i t e coproducts, we w i l l use the notation A.^ + k^ + °*" :+ ^ « n ^ n e coproduct of algebras i s unique up to isomorphism. b) The projection map of X onto X^ provides a natural embedding of A^ i n A. We i d e n t i f y the subalgebras of A so obtained with the A^. Then the A^ form an independent family of subalgebras of A; that i s , for any f i n i t e c o l l e c t i o n of non-zero elements x^ chosen from subalgebras with d i f f e r e n t indices, x^ A X2 A- ... A x R ^ 0. Furthermore, every element of A i s a f i n i t e j o i n of elements of the above form. c) Each i s , i n f a c t , a r e t r a c t of A 0 An easy topological ment y i e l d s a stronger r e s u l t ; i f J i s a subset of I and argu- is a r e t r a c t of A . f o r each j e J , then £(B.; j e J ) i s a r e t r a c t of A. In p a r t i c u l a r , £(AJ: j e J ) i s a r e t r a c t of A„ coproducts are a c t u a l l y contained i n A» We assume these p a r t i a l Under t h i s convention, A i s the union of a l l i t s subalgebras which are f i n i t e coproducts of the A^„ d) If F i i s a closed subset of X^ f o r each i e I, then H(F^t i e I) i s closed i n fl(X^: i e I ) , Proposition 4_: each i e I. J* ± We s p e l l t h i s out a l g e b r a i c a l l y : Let A « Z(A^i i e I), and l e t be an i d e a l of A^ f o r Let J be the ideal of A generated by the union of the Then: i ) "J consists of a l l f i n i t e j o i n s of elements of the form x, A xo A «o« A x , where each x. i s chosen from one of the 1 ^ n i subalgebras A^, and at least one x^ i s chosen from the i d e a l i i ) A/J - L ^ / J ^ : i e I ) . The proof i s a straight-forward v e r i f i c a t i o n . t i o n has some useful The proposi- corollaries: i ) Let I be an ideal of A, J an ideal of B, and l e t K be the i d e a l of A + B consisting of f i n i t e j o i n s of elements of the form a A b, where a E A, b e B and either a e I or b e J . Then A + B/K - A/I + B/J. i i ) If L i s the ideal of A + B consisting of a l l f i n i t e j o i n s of elements of the form a A b where a e l and b e B, then A + B/L * A/I + B. i i i ) If a e A and b e B, then (A + B ) e) - k f l Ab & + B, b If E i s f i n i t e with n atoms, then A + B - A . n §5« ^^hdire^ct Products a) If A = II (A^: i e I ) , then there i s a natural epimorphism of A onto A^ f o r each i £ I. A subalgebra B of A i s c a l l e d a subdirect product of the A^ i f each of these epimorphiems, r e s t r i c t e d to B, s t i l l maps onto A^. b) If { i e 1} i s a c o l l e c t i o n of i d e a l s of an algebra A whose i n t e r s e c t i o n i s the zero i d e a l , then A can be represented as a subd i r e c t "produc t of the 'K/'J^, §6. Free Algebras a) of F i s a free algebra on K generators i f F i s generated by a set X c a r d i n a l i t y K, and any function from X to an algebra A can be extended to a homomorphism of F to A. In case K «* }•{ i s i n f i n i t e , we w i l l denote the free algebra on K generators by F „ a b) of The f r e e algebra on K generators can be r e a l i z e d as the coproduct K copies of the four-element Boolean algebra, or equivalently, as K the algebra of clopen subsets of the Cantor space 2 , the t o p o l o g i c a l product of K copies of the two-element d i s c r e t e space. c) For i n f i n i t e free algebras, the c a r d i n a l i t y of F more, F tion F d) p i s homogeneous and s a t i s f i e s c,c,c. Q is B . a Further- Hence i t s normal comple- i s homogeneous and has c a r d i n a l i t y H ^ , 0 a If A i s a coproduct of K algebras, each with more than two elements, then the free algebra on K generators i s a r e t r a c t of A, e) The countable f r e e algebra F Boolean algebra. q i s the only countable, atomless Its normal completion F i s complete, atomless, and separable, and so i t i s isomorphic to the unique algebra with these properties (see §2, h), f) Free algebras are examples of a more general concept, A Boolean algebra P i s said to be projective i f f o r any homomorphism f of P to 'an algebra *B, '"'ait'd -any "epimorphism P g of A onto B, there i s a homomorphism h of P to A such that A V? B gh = f , §7, Universal Mapping Properties a) Every algebra i s an epimorph of a free algebra. There i s an analogous r e s u l t f o r complete algebras: Proposition 5_: If A i s a complete algebra of c a r d i n a l i t y at most fo^, then A i s an epimorph of F^, Proof: A i s an epimorph of F , which can be embedded i n F , a a Then i n j e c t i v i t y of A y i e l d s the r e s u l t . A form of t h i s r e s u l t was f i r s t proved i n Efimov [10] by a f a i r l y involved topological argument. b) Kausdorff [15] has shown that i f embedded i n P^. = exp/"/ , then Fg can be a This has some important consequences. Proposition 6: _ j — _ Let H g » exp ft . — a Then. F„ 8 i s a subalgebra of a Boolean algebra A i f and only i f P^ i s an epimorph of A. Proof: One d i r e c t i o n follows from the i n j e c t i v i t y of P that i t i s an epimorph of Fg. a and the f a c t The other d i r e c t i o n follows from the -p rQ j ec t i v i t y».a f -« Eg-*- and. ^Hau-sclo r f f •'.s^r-esult. Proposition 7; Proof: that F Since P p Let a exp H ° Then Fg i s a r e t r a c t of P^. a i s complete and Fg i s a subalgebra of i t , we get i s also a subalgebra of i t . But any complete subalgebra of an algebra i s , i n f a c t , a retract of that algebra. c) The most useful form of the preceding r e s u l t s f o r our purposes i s the following: Proposition j j : Let Kg » exp f / . a one another. Then P a and Fg are epimorphs of Corollary; Any two complete algebras of c a r d i n a l i t y exp ft are e p i - morphs of one another. Proof; I t s u f f i c e s to show that i f A i s complete of c a r d i n a l i t y / / exp H then A and P Qt an epimorph of F (§3 8 d) j), P Q are epimorphs of one another. Q arid hence of P , by Prop, 8, Q = By Prop. 5, A i s By Pierce's r e s u l t i s an epimorph of A. If a < £, then P i s a p r i n c i p a l ideal (hence a r e t r a c t ) of Pg, a The algebras are not isomorphic since they do not have the same number of atoms. They might, however, have the same c a r d i n a l i t y . It i s consistent with the usual axioms of set theory to assume, f o r example, that exp ft = exp ft(see Easton [ 9 ] ) . It i s also consistent >to-»assutna--..t'hafc=,tt-< ;-(>.*lmpMes«ex<p-/ty'^ a quence of GCH, Proposition 9; t i f exp H a Proof: In any case, we have the following: P a and P^ are epimorphs of one another i f and only = exp ft^ 0 Assume exp ft a a exp ft^ •= H^* By Prop. 8» P morphs of one another, as are Pg and F^, algebras are epimorphs of one another. e) a and F^ are e p i - Hence, the two power-set The other d i r e c t i o n i s c l e a r , Pierce's r e s u l t (§3, j ) has u s e f u l consequences. If A i s an i n f i n - i t e epimorph of a o-complete algebra, then A has no i n f i n i t e f r e e e p i morphs. In p a r t i c u l a r , no i n f i n i t e complete algebra can have an i n - f i n i t e free epimorph. CHAPTER ONE PADICAL CLASSES OF BOOLEAN ALGEBRAS The concept of a r a d i c a l c l a s s of rings i s veil-known and has been studied extensively. The theory can be applied immediately to the c l a s s of Boolean r i n g s , but some adjustment i n the d e f i n i t i o n s and r e s u l t s i s necessary f o r the c l a s s of Boolean algebras, §1, General R.adical Theory T h i s section i s a review of the basic concepts of r a d i c a l •'theory, f orv-sssociaftive* rings, • - A 'general '.reference «f or t h i s material i s Divinsky [8], 1.1 D e f i n i t i o n : A c l a s s of associative rings i s c a l l e d u n i v e r s a l i f i t i s closed under the formation of epimorphs and i d e a l s . In what follows, we assume that a l l classes of rings considered are subclasses of some f i x e d u n i v e r s a l c l a s s , 1.2 D e f i n i t i o n (Amitsur [2], Kurosh [17]): A non-empty c l a s s R of rings i s a r a d i c a l c l a s s (or simply a r a d i c a l ) i f i t s a t i s f i e s the following properties; i ) every epimorph of an R-ring i s an R-ring, i i ) every r i n g A contains an R-ideal, c a l l e d the R-radical of A and denoted by r ( A ) , which contains every R-ideal of A, and i i i ) f o r any r i n g A, r(A/r(A)) = 0. 1„3 Definition; If R i s a r a d i c a l c l a s s and A i s a r i n g such that r(A) » 0, then A i s c a l l e d R-seal-simple. When some fixed r a d i c a l class i s being discussed and there i s no danger of ambiguity,, we w i l l simply use the terms " r a d i c a l " and "semi-simple" without s p e c i f i c reference to the r a d i c a l class,, It i s obvious that the t r i v i a l ring {0} i s the only r i n g which can be simultaneously radical 1.4 r a d i c a l and semi-simple with respect to a class. Proposition; For any r a d i c a l class R and any r i n g A, r(A) i s the i n t e r s e c t i o n of a l l i d e a l s I of A f o r which A/I i s R-semi-simple. The following propositions characterize r a d i c a l c l a s s e s and give some of t h e i r closure properties. 1.5 Proposition; A c l a s s R i s a r a d i c a l class i f and only i f : i ) R i s closed under epimorphs, and i i ) i f A i s a r i n g such that every non-zero epimorph of A has a non-zero R-ideal, then A i s i n R. ^••6 Proposition; If R i s a r a d i c a l c l a s s and I i s an ideal of a ring A which i s generated by R-ideals of A, then I i s an R-ideal Of A 1.7 e Corollary; If R i s a r a d i c a l c l a s s , then the weak d i r e c t product of R-rings i s an R-ring, 1.8 Proposition: If R i s a r a d i c a l c l a s s and I i s an R-ideal of a ring A such that A/I i s i n R, then A i s i n R, 1.9 Def i n i t i o n : For any c l a s s M of r i n g s , we say a r i n g A i s an approximate W-ring^ i f every non-zero i d e a l of A has a non-zero e p i morph i n M, 1.10 , Proposition; A c l a s s l o f .rings M is. .thexlass of . a l l R-semi- simple r i n g s f o r some r a d i c a l class R i f and only If M i s equal to the class of a l l approximate M-rings. In t h i s case, R can be recov- ered from M as the c l a s s of a l l rings with no non-zero epimorph i n M, 1.11 Def i n i t i o n ; A class of rings i s c a l l e d hereditary i f every . ideal.of a r i n g i n the c l a s s i s also i n the c l a s s , 1.12 Proposition (Armanderiz [3]): A class M i s the c l a s s of a l l semi-simple rings f o r some r a d i c a l c l a s s i f and only i f : i ) W i s hereditary, i i ) M i s closed under subdirect products, i i i ) i f I i s an M-ideal of a r i n g A for which A/I i s i n M, then A i s i n M, and i v ) i f I i s an ideal of a r i n g A such that I/B i s a nonzero M-ririg f o r some ideal B.of contained I, then there i s an i d e a l C of A i n I such that I/C i s a non-zero M-ring. The f a c t that any semi-simple c l a s s of a s s o c i a t i v e r i n g s i s hereditary was 1.13 IH f i r s t proved by Anderson, Divinsky, and Proposition; S u l i n s k i [1], A r a d i c a l R i s hereditary i f and only i f r ( I ) r(A) f o r any r i n g A and any i d e a l I of A. Furthermore, i f R is hereditary, then f o r any r i n g A, r(A) i s the i d e a l of A generated by the p r i n c i p a l R-ideals of 1.14 Definition; A 0 For any c l a s s C of rings and any r i n g A, define c(A) to be the ideal of A generated by the p r i n c i p a l C-ideals of A. This coincides with the d e f i n i t i o n of r(A) f o r a r a d i c a l c l a s s R provided that R i s hereditary. In the ©ext section, we show that a l l r a d i c a l s we consider are hereditary, so the notation w i l l be unambiguous. §2„ Radical Classes of Boolean Rings The class of Boolean rings i s a u n i v e r s a l class, so we can immediately apply the concepts and r e s u l t s of the previous section. We w i l l prove only what i s needed to f a c i l i t a t e the passage to Boolean algebras. 1.15 Proposition: If R i s a non-zero r a d i c a l c l a s s of Boolean r i n g s , then the two-element Boolean algebra 2^ i s i n R. Proof: 1.16 Any non-zero Boolean r i n g has 2_ as an epimorph. Corollary; If R i s a non-zero r a d i c a l c l a s s of Boolean rings, and A i s a Boolean r i n g with a maximal i d e a l I i n R, then A i s i n R, Proof: 1»17 Both I and A/I - 2_ are in R. Proposition: By Prop, 1.8, A i s i n R. Every r a d i c a l c l a s s of Boolean rings i s hereditary. Proof: Let A be i n the r a d i c a l c l a s s R of Boolean r i n g s . I of A i s generated by the p r i n c i p a l i d e a l s of A contained Any i d e a l within i t . "Each o f ' t h e s e ' i s an epimorph b'f 'A and so'is'in'R, ""By'Pro'p. '1,*'6, I i s i n R. We are now ready to prove the theorem which allows us to r e s t r i c t attention to Boolean algebras, 1.18 Theorem: Let R be a non-zero r a d i c a l c l a s s of Boolean r i n g s , and l e t S ° A fi R, the c l a s s of Boolean algebras i n R, Then f o r any Boolean r i n g B, the following are equivalent: i ) B i s i n R, i i ) every non-zero p r i n c i p a l i d e a l of B i s i n 5, i i i ) . . C i s i n S, where C i s the Boolean algebra B as a maximal i d e a l . containing Froof; If B i s i n R, then every non-zero p r i n c i p a l of B i s i n R by Prop. 1.17,, and C i s in R by Cor. 1.16. are, i n f a c t , i n S. Being algebras, they Thus i ) implies i i ) and i i i ) . then every p r i n c i p a l i d e a l of B i s in R. If i i ) holds, Since B i s generated by i t s p r i n c i p a l ideas, i t i s i n R by Prop. 1.6. Thus i i ) implies i ) . If i i i ) holds, then C i s i n R, so by Prop. 1,17, B i s in R. So i l l ) implies i ) , §3, Radical Classes of Boolean Algebras 1.19 a Definition; A non-empty class 5 of Boolean algebras i s c a l l e d r a d i c a l class i f and only i f there i s a r a d i c a l class R of Boolean rings such that S = A ft R, The remainder of t h i s section shows that a l l the concepts and results of r a d i c a l theory can be expressed (with only minor modi- f i c a t i o n s ) e n t i r e l y i n the language of Boolean algebras. with Defn. 1,14, In keeping f o r any c l a s s S of algebras, the ideal s(A) of the algebra A i s the ideal generated by the S-ideals (necessarily p r i n c i p a l ) of A, 1.20 Proposition: A class S of Boolean algebras i s a r a d i c a l c l a s s i f and only i f : i ) S i s closed under algebra epimorphs, and i i ) s(A/s(A)) = 0 f o r every Boolean algebra A. Proof: If S i s a r a d i c a l c l a s s of algebras obtained from the Boolean ring r a d i c a l R, then i t i s clear that s(A) «* r(A) f o r any algebra A, Hence, i i ) follows immediately. Also, any algebra eplmorphism i s a ring epimorphism, so i ) follows. Suppose S s a t i s f i e s i ) and i i ) , and l e t R be the c l a s s of a l l Boolean rings A s a t i s f y i n g A j . e S f o r every non-zero x e A, It i s c l e a r that S i s p r e c i s e l y the class of a l l algebras i n R, so a l l need show i s that R i s a r a d i c a l c l a s s of rings. ring epimorph of the R-ring A, we Suppose that B i s a Then any non-zero p r i n c i p a l i d e a l of B i s a r i n g epimorph of some non-zero p r i n c i p a l i d e a l of A, which i s an 5-algebra. Moreover, a non-zero ring eplmorphism on an algebra must preserve the unity and so i s , i n f a c t , an algebra eplmorphism. By I ) , then, every non-zero .principal ideal of B i s an .S-algebra, and so K B e R. Now suppose A i s a ring such that every non-zero epimorph of A lias a non-zero R-ideal. If A i s not i n R, then there i s a non-zero x e A such that A^ i s not i n S, But then A^/s (Aj,) i s a non-zero ring epimorph of A and so must have a non-zero R-ideal. But then the algebra A / s ( A ) has an S - i d e a l , contradicting condition x x ii). By Prop, 1.5, R i s a r a d i c a l class of Boolean rings. 1»21 Proposition: A c l a s s 5 of Boolean algebras i s a r a d i c a l c l a s s i f and only i f : i ) S i s closed under algebra epimorphs, and i i ) i f A i s an algebra such that every algebra epimorph of A has an S-ideal, then A i s i n 5. ' Proof: Suppose S i s a class s a t i s f y i n g i ) and i i ) , define R as i n the l a s t proposition, and repeat the argument showing that R i s closed under ring epimorphs. Now l e t A be a r i n g such that every non-zero ring epimorph of A has a non-zero R-ideal. In p a r t i c u l a r , every non-zero. • . ..._ ---j.- _ p r i n c i p a l i d e a l of A s a t i s f i e s t h i s condition, and so has an S - i d e a l . By i i ) , the, every non-zero p r i n c i p a l i d e a l of A i s i n 5, and so A i s in R. Thus, R i s a r a d i c a l class of Boolean rings and S consists of a l l the algebras 1.22 in i t . Proposition; The opposite d i r e c t i o n i s c l e a r . Let S be a r a d i c a l c l a s s of algebras, from the Boolean r i n g r a d i c a l R, Then for any algebra obtained (ring) A; s(A) = r ( A ) «= {x: A, e R} » {x; x = 0 or A„ e S}„ x x ..Proof: ...Let..I « {.xs.-A~e*-R} , ..,A11 that,.needs,proof i s that X i s an x ideal of A. Since R i s hereditary, I i s closed under subelements. The f a c t that I i s closed under f i n i t e j o i n s follows from the f a c t that A xvy i s generated by A and A , and Prop, 1.6, x y* 1 This proposition makes precise what we w i l l mean by the expression: the r a d i c a l of an algebra c o n s i s t s of i t s r a d i c a l elements. 1.23 Definition; If S i s a r a d i c a l c l a s s of algebras, we say an algebra A i s S-semi-simple i f s(A) » 0. A c l a s s of algebras i s a_ semi- simple c l a s s i f i t consists of a l l S-semi-simple algebras f o r some radical c l a s s S of algebras. •• . 1.24 Definition; For any c l a s s M of algebras, we say an algebra A i s an approximate M-algebra i f every non-zero p r i n c i p a l i d e a l of A has an (algebra) epimorph i n M. 1.25 A class of algebras U i s a semi-simple c l a s s i f Proposition; and only i f M i s equal to the c l a s s of a l l approximate M-algebras. In t h i s case, the r a d i c a l class associated with M i s the class of a l l algebras with no (algebra) epimorph i n M. Proof: A straight-forward v e r i f i c a t i o n , similar i n s p i r i t to Prop, 1.20 and Prop, 1.21. §4, Conventions and Summary Unless otherwise stated, we w i l l henceforth refer only to classes of algebras. The d e f i n i t i o n s of c l a s s properties w i l l be modified i n the obvious manner. 1.26 Definition: For example: A class of algebras i s said to be hereditary i f i t Is closed under the formation of non-zero p r i n c i p a l i d e a l s . The term "epimorph" w i l l henceforth mean "algebra epimorph," We r e t r i e v e the symbol R for a r a d i c a l class of Boolean algebras. We make the convention that an i d e a l I of an algebra A w i l l be c a l l e d an R-ideal of A i f A need not be i n R, x e R for every x e I, Hence an R-ideal The following propositions summarize the properties of r a d i c a l and semi-simple classes which we w i l l find most u s e f u l . Some of these properties have already been proved, and the others are straight-forward extensions of known r e s u l t s , •*-»27 Proposition; i) 2 e Let R be a r a d i c a l class of Boolean algebras. R Then: p i i ) R i s hereditary, i i i ) i f A i s an algebra with an R-ideal I such that A/I i s i n R, then A i s i n R, i v ) i f I i s an ideal of an algebra A generated by R-ideals of A, then I i s an R-ideal of A, v) R i s closed under the formation 1,28 Then: Proposition: of weak products, Let M be a semi-simple c l a s s of Boolean algebras. i ) M i s hereditary, i i ) M i s closed under subdirect products, i i i ) i f I i s an M-ideal of an algebra A such that A/I i s i n W, then A i s i n M, CHAPTER TWO THE LOWER RADICAL Given any c l a s s of r i n g s , i t i s possible to construct a smallest r a d i c a l containing t h i s c l a s s (see Divinsky [8]). We Investigate the s p e c i a l features of the general construction i n the case of Boolean algebra radicals, §1, The Lower Radical Construction 2,1 PH^Jgosl^ion. of r i n g s . Let X be any c l a s s of rings i n some u n i v e r s a l c l a s s For any o r d i n a l a, we define a c l a s s X^ as follows: i) x o - X, i i ) X^ i s the class of epimorphs of X-rings, i i i ) f o r a > 1, assuming X^ has been defined f o r a l l 3 < a, let X a be the c l a s s of a l l rings A such that every non-zero epimorph of A has a non-zero i d e a l belonging to X^ f o r some B < a. Let L(X) be the union of the classes X „ a radical class. Then L(X) i s a Furthermore, i f R i s a r a d i c a l c l a s s containing X, then R contains L(X)„ If H i s a c l a s s of rings closed under epimorphs and R = L(H), we have a construction due to Amitsur [2] which, f o r any r i n g A, y i e l d s an i d e a l h*(A) h*(A) of A. Whenever H s a t i s f i e s an a d d i t i o n a l condition, coincides with r (A) , thus giving an i n t e r n a l . I t e r a t i v e construc- t i o n of the r a d i c a l of A i n terms of 2.2 H, D e f i n i t i o n (Pierce): A ladder i n a ring A i s a chain of ideals of A, 0 <= I « 1^ $ ... < * Q a well-ordered » ^ s u c n that i f a i s a l i m i t o r d i n a l , then I i s the union of the I f o r 3 < a. We " a 3 note that there i s a l e a s t o r d i n a l 6 such that I = I for a l l a £ a o and we c a l l Ig the summit of the ladder { l K n s 6, * a 2.3 Lemma: Let R be a r a d i c a l c l a s s and suppose { I J i s a ladder i n a a r i n g A with summit Ir, s a t i s f y i n g I ,,/! e R f o r a l l o r d i n a l s a, o" . a+1 a Then 1 i s in R. & "Proof: ' U s i n g P r o p . 1.6 I Q 2.4 and'Prop, 1.8, an easy induction shows that i s i n R for a l l ordinals a, Definition: Let H be a c l a s s of rings closed under epiroorphs. For any r i n g A, we define a ladder i n A as follows: i ) h (A) Q - 0 . i i ) assuming that h (A) Q has been defined, h ^ ( A ) / h (A) i s +1 the i d e a l of A/h (A) generated by I t s H-ideals, a i i i ) i f a i s a l i m i t o r d i n a l and 6 < a, then h (A) o We by h*(A). hg(A) i s the union of the hg(A) and has been defined f o r a l l f o r 3 < a. c a l l {1^(A)} the tf-ladder i n A and denote i t s summit Using Prop. 1.6 2.5 Lemma: S we e a s i l y get: If H i s closed under epimorphs and R » L(H), then f o r a l l rings A, h*(A) « r ( A ) . 2.6 Def i n i t i o n : A c l a s s of rings tf, closed under epimorphs, i s c a l l e d an Amitsur c l a s s i f h*(A) = 0 implies h*(I) = 0 f o r every ideal I of A. 2.7 Proposition: Let If be closed under epimorphs and l e t R «= L(H). Then r(A) « h*(A) f o r a l l rings A i f and only i f H i s an Amitsur c l a s s . The proof of the s u f f i c i e n c y i s due to Amitsur [2], The necessity follows from the fact that semi-simple classes are hereditary. §2, The Lower Radical f o r Eoolean Algebra Radicals If X i s a c l a s s of Boolean algebras, we can use i t to generate a Boolean r i n g r a d i c a l , and thence a r a d i c a l c l a s s of algebras L(X) which i s minimal with respect to containing X, This assures us that there i s a lower r a d i c a l construction f o r Eoolean algebra r a d i c a l s . Using the fact that i d e a l s of i d e a l s of an algebra are themselves i d e a l s of that algebra, we can show that the general construction of Prop, 2,1 stops a t the second stage f o r Boolean algebra r a d i c a l s . 2.8 Proposition: Let X be any class of Boolean algebras, and l e t R be the class of a l l algebras A such that every epimorph of A has an i d e a l (necessarily non-zero p r i n c i p a l ) which i s an epimorph of an X-algebra. Then R » JL(X), Proof: Since X S R G L ( X ) , i t s u f f i c e s to show that R i s a r a d i c a l class. C l e a r l y , R i s closed under epimorphs. Now suppose that A i s an algebra such that every epimorph of A has an i d e a l i n R, show that A i s i n R, B has an i d e a l C i n R, Suppose B i s an epimorph of A, By We must . assumption, Then C, being an epimorph of i t s e l f and i n R, contains an i d e a l D which i s an epimorph of an X-algebra, But then D i s an i d e a l of B, and so every epimorph B of A contains an i d e a l which i s an epimorph of an X-algebra. Thus A i s i n R as required, and R i s a r a d i c a l cTas's, The following d e f i n i t i o n s and lemma w i l l be u s e f u l i n determining the structure of algebras i n a lower r a d i c a l , p a r t i c u l a r l y when the generating c l a s s i s closed under epimorphs extremely u s e f u l i n Chapter F i v e . 9 and they w i l l also be F i r s t , r e c a l l (Preliminaries, §2, f ) that any algebra has a product decomposition across any element and i t s complement. More generally, i f D i s any d i s j o i n t e d subset of an algeb- ra A with sup D ra 1, then A has a dense ideal-generated subalgebra i s o - morphic to wH(A^: d e D), When A i s complete, we have A - lUA^: d e D), We extend these r e s u l t s as follows: 2,9 Definition: If P i s a property of algebras, we say x Is a P-element of A i f A i f , whenever x i s a P-algebra. The property P i s hereditary y are elements of A and y i s a P-element of A, then x i s also a P-element of A, Whenever an algebra A has a dense subset of P-elements, f o r some hereditary property P$ then any maximal d i s j o i n t subset D of P-elements must s a t i s f y sup D ™ 1, 2»1Q Lemma: Hence, we e a s i l y get the following: Let P be a hereditary property of Boolean algebras. If A i s an algebra with a dense subset of P-elements, then A lias a dense ideal-generated subalgebra isomorphic to a weak product of P-algebras, If A Is a complete algebra, then A i s a product of P-algebras i f and only i f the P-elements of A are dense i n A, " Noting that the property of being a two-element algebra i s hereditary, we immediately deduce the following well-known r e s u l t : 2„ 13. Corollary: A complete algebra i s a product of two-element algeb- ras (that i s , a power-set algebra) i f and only i f i t i s atomic (that i s , the atoms are dense in, the algebra), 2,12 theorem: Suppose H i s a class of algebras closed under epimorphs, and l e t R =» L(H), Then: i ) A i s i n R i f and only i f every epimorph of A has an i d e a l i n H, i i ) A i s i n R i f and only i f every epimorph of A has a dense subset of H-elements, i i i ) any R-algebra A has a dense ideal-generated subalgebra isomorphic to a weak product of H-algebras, i v ) any. complete algebra A i n R i s a product of H-algebras, v) R contains atomless complete algebras i f and only i f H does. Proof; The f i r s t assertion follows immediately from Prop, 2 8, 0 Since p r i n c i p a l i d e a l of an algebra are epimorphs of that algebra, i i ) follow from i ) . Using Lemma 2,10, we immediately get i i i ) and i v ) from i i ) . and v) follows from i v ) , 2.13 Proposition; Let R = L(H) as i n Prop. 2.12, and l e t A be a homo- geneous algebra* Proof; Then A i s i n R i f and only i f A i s i n H, If A i s i n R, then every epimorph of A and i n p a r t i c u l a r , A i t s e l f , has an i d e a l i n tf. Since A i s isomorphic to any p r i n c i p a l i d e a l of i t s e l f , A i s i n K, The other d i r e c t i o n i s obvious. Homogeneous algebras are a special case of the following more general concept: 2.14 for D e f i n i t i o n (Divinsky): A Boolean algebra A i s unequivocal i f every r a d i c a l class R, A i s either i n R or i s R-semi~simple, 2 15 0 Proposition: An algebra A i s unequivocal i f and only i f L (AJJ) = L (A) f o r every non-zero x i n A, Proof; We always have L(A ) contained i n L(A), If A i s unequivocal, x then i t must be i n L(A^) since i t cannot be semi-simple with respect to t h i s r a d i c a l . Hence we get L(A) = L(A ) f o r any non-zero x i n A, Conversely, suppose t h i s always holds f o r an algebra A, and suppose that A i s not R-semi-simple, f o r some r a d i c a l R» zero x i n A p A^ i s i n R. and so A must be i n R. 2.16 Corollary: But then, L (A) = *-( ) A x Then f o r some noni s contained i n R, Hence A i s unequivocal. Every non-zero p r i n c i p a l i d e a l of an unequivocal algebra i s an unequivocal algebra. •^2"*3f-7 '^ExaB^ • -for example, l e t A be the product of non-isomorphic homogeneous algebras A-^ and A^, which are epimorphs of one another. Such p a i r s e x i s t : take A, = F and A„ «= P / I , where I i s the ideal of f i n i t e sets i n P . i o 2 o " o C l e a r l y , i n any such s i t u a t i o n , L (A) = L(A^) = 1 ^ 2 ) . Now l e t x = ( x , X£) be a non-zero element of A, so that either x^ or x^ I s non± zero. Then A has a p r i n c i p a l i d e a l , hence an epimorph, isomorphic to either A^ or A£. So 1 ( A ) = 1(A) f o r a l l non-zero x i n A, and A X i s unequivocal. We now show that the Aoitsur procedure f o r obtaining the r a d i c a l i s always v a l i d f o r Boolean algebras. 2.18 Proposition: If H i s a c l a s s of Eoolean algebras closed under epimorphs, then H i s an Amitsur c l a s s . Proof: Suppose that A i s an algebra f o r which h*(A) = 0 and that I i s an i d e a l of A, If I has any ideals in ft, then A has these same ideals i n H, contradicting h*(A) = 0, Thus, h*(I) «• 0, and H i s an Amitsur c l a s s by Defn, 2,6, §3, The Superatomic 2.19 Radical D e f i n i t i o n (Mostowskl and T a r s k i [21]): A Eoolean algebra i s said to be superatomic i f every epimorph has an atom. The concept of a superatomic Eoolean algebra was first .->;S£ud±edi\by .Kostcwek^^ Day [6], [7], The following proposition from Day various characterizations of superatomic Boolean 2.20 Proposition: If A i s a Boolean algebra, [7] summarizes the algebras, then the following are equivalent: i ) A i s superatomic, i i ) every epimorph of A i s atomic, i i i ) every subalgebra of A has an atom, i v ) every subalgebra of A i s atomic, v) no subalgebra of A i s an i n f i n i t e free algebra, v i ) A has no subalgebra isomorphic to F , v i i ) A has no chain of elements order-isomorphic with the chain of r a t i o n a l numbers, v i i i ) the Stone space of A i s clairseme; that i s , every nonempty subspace of S(A) has an isolated point. If we l e t H be the class consisting of the two-element algebra, then H i s closed under epimorphs, and Defn. 2.19 can c l e a r l y be re-stated as follows: A i s superatoiaic i f and only i f every e p i morph of A has an i d e a l i n ff. 2,21 Proposition: By Theorem 2.12, The c l a s s 0 of superatomic Boolean algebras i s a r a d i c a l class; namely, i t i s L(2), two-element 2o*22 the lower r a d i c a l generated by the Boolean algebra, Corollary: Every r a d i c a l c l a s s of Boolean algebras contains 0, Every r a d i c a l class of Boolean algebras contains J2. Proof: 2.23 we have: Corollary (Day [7]): The weak product of superatomic Boolean algebras i s a superatomic Boolean algebra. By Prop. 2,18, we can use Amitsur*s constuction to obtain the superatomic r a d i c a l of any algebra. 2.24 Proposition: Then h Let H =» {2} The following i s c l e a r : and l e t A be any Eoolean algebra. ^ (A)/h^(A) i s the Ideal of A/h (A) generated by i t s atoms; o more p r e c i s e l y , i t consists of a l l f i n i t e j o i n s of atoms of A / h ( A ) . a If A i s superatomic, then i t i s the summit of i t s H-ladder. Let 6 be the l e a s t o r d i n a l f o r which h.(A) = A, 6 2.25 D e f i n i t i o n (Day [7}): The c a r d i n a l sequence of the superatomic Boolean algebra A i s the sequence of order type 6 whose a-term, f o r a < 6, i s the c a r d i n a l i t y of the set of atoms of A/h (A), a This s p e c i a l i z a t i o n of the Amitsur construction has been used with success by Day i n h i s study of superatomic algebras. We mention two of h i s more s t r i k i n g r e s u l t s . 2.26 Proposition (Day [ 7 ] ) : Two countable superatomic Boolean a l g - ebras are isomorphic i f and only i f they have the same cardinal sequence, 2.27 Proposition (Day [7]): If k i s an i n f i n i t e c a r d i n a l , then there are more than K non-isomorphic superatomic Boolean algebras of cardina l i t y K„" Superatomic Boolean algebras have arisen n a t u r a l l y i n the study of free complete extensions of an algebra. 2.28 Definition: C i s a free complete extension of B i f : i ) C i s complete, i i ) B i s a subalgebra of C, i i i ) any homomorphism of B to a complete algebra can be extended to a complete homomorphism (that i s , one which preserves a l l suprema) of C to that algebra. If B has a f r e e complete extension, then i t i s unique up to isomorphism, 2.29 Proposition (Yaqub [27], Day [ 6 ] ) : B has a f r e e complete extension i f and only i f B i s superatomic. Further r e s u l t s of a s i m i l a r nature occur i n these papers. The C a r d i n a l i t y Radicals 2.30 Definition: For any o r d i n a l a, we c a l l the lower r a d i c a l generated by F , the free algebra on H a a generators, a c a r d i n a l i t y r a d i c a l and denote i t by F « 2.31 of Proposition; F ^ i s the lower r a d i c a l generated by the c l a s s Boolean algebras of c a r d i n a l i t y at most H » Moreover „ i f o < a then F Proof; of A i s properly contained i n F ^ . The f i r s t a s s e r t i o n follows from the f a c t that every algebra c a r d i n a l i t y at most H a epimorph of F g S i s an epimorph of F , a so we get F ^ contained i n F^, If o < &, F i s an * a But F^ i s i n F ^ and not i n F „ so the containment i s proper. A In a natural sense B the superatomic r a d i c a l i s the f i r s t member of t h i s chain, f o r i t i s the. lower r a d i c a l generated by a l l f i n i t e Boolean algebras. We s h a l l see i n Chapter Seven that F q i s an atom i n the l a t t i c e of Boolean algebra r a d i c a l s , so we have some i n t e r e s t i n examples and some properties of F - a l g e b r a s . o 2,32 Examples; in F . Of course, any weak product of countable algebras i s Furthermore, any atomless algebra i n F generated subalgebra Q has a dense i d e a l - isomorphic to a weak product of copies of F . Q The normal completion of any atomless algebra i n F Q i s a power of F , Q and any power can be so r e a l i z e d ; that i s , as the normal completion of an F ^ - algebra. Let S be a superatomic algebra with K atoms. Since any element i n an atomic algebra i s the supremum of the atoms below i t , we can embed S i n P(tc), which can be r e a l i z e d as the subalgebra of elements with {0, l}-coordinates i n any K-product. can assume that P ( K ) i s embedded i n F generated by the atoms of S, F Q q Let I be the i d e a l of F we Q Then any non-zero p r i n c i p a l i d e a l of contained i n I i s isomorphic to F . Q algebra of F^ , In p a r t i c u l a r , Furthermore, i f A i s the sub- generated by S and I, then I i s also an i d e a l of A, and A/I i s superatomic. In f a c t , A/I i s isomorphic to the quotient of S by the i d e a l generated by i t s atoms. Prop, 2,22, A i s i n F , q Hence, by Prop. 1.6 and I n t u i t i v e l y , we can regard t h i s as the replacement of each atom i n S by a copy of F . Q In Chapter Six, we w i l l describe another method of o b t a i n ing F -algebras from superatomic algebras. A natural extension of countable Boolean algebras are the separable algebras, that i s , those having a countable, dense subset,, 2.33 Def i n i t i o n : K i s the lower r a d i c a l generated by the separable Boolean algebras. We w i l l f i n d use f o r t h i s r a d i c a l i n Chapter Seven. 55. The Power-Set Radicals 2.34 Definition: The lower r a d i c a l generated by P , the algebra of a a l l subsets of a set of c a r d i n a l i t y # , w i l l be c a l l e d a power-set a r a d i c a l , and denoted by P .. 2.35 P r o p o s i t i o n : Let H - exp ft . -— 3 'a D Proof: 2.36 By Prop. 0.8, P Proposition: more, ? Proof: a Q Then P a - L(F„). (3 and Fg are epimorphs of one If a ^ 6, then ? - Pg i f and only i f exp ft a a a another. i s contained i n Pg. Further- exp ftg. The f i r s t assertion follows from the fact that P i s an e p i - morph of Pg, and the second from the equivalence of the c a r d i n a l i t y condition with the f a c t that P using Prop. 2.37 and Pg are epimorphs of one Q another, 0.9. Theorem: Let ft « exp ft , Q Then the following are equivalent: Let A be a complete algebra. i) A ? , e a i i ) A i s an epimorph of P f i i i ) A i s an epimorph of F^, iv) Proofs |A| Using Prop, 0 5 and Prop, 0.8 C we see that i i ) i i i ) and i v ) 0 e are equivalent, and they c l e a r l y imply i)» Now assume i ) . Then by Theorem 2.12, A i s a product of epinorphs of P , Q Let 0 H Q » T h e n of P „ each a Pg can be represented as & product of of which has one of the A^ as an epimorph. their product A as an epimorph, morph of P . say A = H(A^; We must now |l|-copies Then Pg has show that Pg i s an e p i - T h i s i s obvious i f 8 < a so assume a < 8. Since H a K m D a max (H t i e 1). p t h i s means that = Since A i s i n P , and has Q P(1) »"Fg as a r e t r a c t , t h i s iiieans that "is i n P »"hence that 0 « P . ... But t h i s i s equivalent to the f a c t that P a a and Pg are e p i - morphs of one another. 2.38 ? o Corollary; Let A be an i n f i n i t e complete algebra. i f and only i f |A| « exp HQ* In t h i s case, L(A) » P Q Then A i s i n and i f A i s atomless, then i t i s unequivocal. Proof; Any i n f i n i t e complete algebra has c a r d i n a l i t y at l e a s t exp $i , Q The rest follows from the C o r o l l a r y to Prop. 0.8 and from Prop. It i s i n t e r e s t i n g 2,15. to note, here, that Monk and Solovay [20] have shown that there are exp exp H - isomorphism classes of complete a Boolean algebras of c a r d i n a l i t y exp H. a We w i l l be interested i n Chapter Five with those complete algebras that generate a power-set 2.39 Theorem; Let = exp f ^ . radical, Let A be any algebra i n ? . Then a the following are equivalent; i ) L(A) ii) P a P A P i s an epimorph of A, i i i ) Fg i s an epimorph of A, i v ) Fg i s a subalgebra of A. If A i s any complete algebra, then A generates P r a d i c a l p r e c i s l y when i t has c a r d i n a l i t y q f o r i t s lower and s a t i s f i e s any one of i i ) , i i i ) or i v ) , which i s equivalent to saying that A and ? a are e p i - morphs of one another. Proof; Using Prop, 0.6 and Prop. 0.6, we see that i i ) , i i i ) and i v ) are equivalent to one another, and i t i s clear that they imply i ) . Assuming i ) , we see that F^ i s i n L (A), epimorph of A by Prop. 2.13. algebra, then A e ? a Being homogeneous, i t i s an Thus i ) implies i i i ) . If A i s a complete i s equivalent to the inequality |A| S W^e which, i n the presence of any of the conditions i ) - i v ) , must a c t u a l l y be an e q u a l i t y . The rest follows immediately. F i n a l l y , we introduce a related r a d i c a l ; 2.40 Definition; P i s the lower r a d i c a l generated by the c l a s s of a l l complete algebras. 2.41 Proposition; P i s the lower r a d i c a l generated by a l l power- set algebras, and so i s the smalle st r a d i c a l c l a s s containing a l l the P .. Proof: O Every complete algebra i s a r e t r a c t of a power-set algebra CHAPTER THREE THE UPPER RADICAL Given any class of rings s a t i s f y i n g a condition called r e g u l a r i t y , i t i s possible to generate a semi-simple class which i s minimal with respect to containing the given class,, §1. The Uffjagr Radical Construction We describe the construction immediately f o r classes of Boolean a l g e b r a s 0 The method and proof of i t s v a l i d i t y are esaen- . •"•~'CMl*ly^the-'Baoe"a»'in*general"'radical"theory "(see•Divinslcy" [8]-) "with minor obvious modifications. We r e c a l l that the two-element algebra must be i n any radical class. This e n t a i l s that semi-simple algebras must always be atomless, 3,1 Definition: A c l a s s H of Boolean algebras w i l l be c a l l e d regula i f every M-algebras i s atomless, and every algebra i s an approximat M-algebra (Defn. 1.24), Of course, any hereditary class of atomless algebras i s a regular c l a s s . 3.2 Proposition: Let M be a regular class of Boolean algebras, and l e t U(M) denote the class of a l l Eoolean algebras with no epimorph i n M. Then: i ) U(M) i s a r a d i c a l class of algebras, i i ) every algebra i n W i s semi-simple with respect to U(M), i i i ) i f R i s a r a d i c a l c l a s s such that every algebra i n M i s R-semi-simple, then R i s contained i n U(M). 3.3 Definition: The r a d i c a l class U(M) determined by the c l a s s i s c a l l e d the upper r a d i c a l W, Given any c l a s s V of atomless Boolean algebras, i t i s c l e a r that the class U of a l l non-zero p r i n c i p a l i d e a l s of y-algebras i s a ,her.e,d i t a r - y c l a s s. ...o f „ .atomle ss, algebras.. 3.4 Definition: If y i s any c l a s s of atomless Boolean algebras and M i s the class of a l l non-zero p r i n c i p a l ideals of ^-algebras, then we w i l l denote the r a d i c a l c l a s s ti(M) by U(/) and c a l l i t the upper r a d i c a l determined by the c l a s s V» §2. The Characterization 3.5 Theorem: Theorem Let M be a regular c l a s s of Boolean algebras, and l e t U(M) be the upper r a d i c a l determined by M. U(M)-semi-simple Then an algebra A i s i f and only i f i t i s a subdirect product of M-algebras, Proof; By Prop. 1.28,, semi-simple classes are closed under subdirect products, so one d i r e c t i o n i s immediate. The other d i r e c t i o n w i l l follow i f we can show that f o r R » U(W)« and f o r any Boolean algebra A, r(A) i s equal to the intersection J of a l l i d e a l s I of A such that A/I i s i n W. Then, for a semi-simple algebra, t h i s i n t e r s e c t i o n w i l l be 0, and A w i l l be a subdirect product of the A/I. By Prop. 1.4, r(A) i s the intersection of a l l Ideals I of A for which A/I i s R-semi-simple, and so i t i s contained i n J , If they are not equal, l e t x be an element of J which i s not i n r (A). i s not i n R, and so has an epimorph A / K i d e a l of A. in Then A^ Note that K i s also an Let L denote the ideal of A generated by K. and x*. using §2, f ) of the Preliminaries, we see that A/L - A^/K, Then, which i s i n M. *lience L i s * one* of'the "ideals occurring"in the def i n i t i o n of J , and so J i s contained i n L« But then x must be i n L and, being d i s j o i n t from x', i t must be i n K, But t h i s y i e l d s the contradiction A^/K = 0. Hence J » r(A) as required, ^ •' Atomless Boolean Algebras The superatomic r a d i c a l 0 Is c l e a r l y the upper r a d i c a l determined by the atomless Boolean algebras. We f i n d a much smaller regular c l a s s which determines 0 as i t s upper r a d i c a l . 3«6 Theorem; algebras. Let M be the c l a s s of a l l atomless, separable Boolean Then: i ) M i s a hereditary c l a s s , and so determines an upper radical, i i ) t h i s r a d i c a l i s , i n f a c t , the superatomic r a d i c a l , and so i i i ) an algebra A i s atomless i f and only i f i t i s a subdirect product of atomless, separable algebras. Proof; It i s c l e a r that M i s hereditary. By Prop. 2,20, an algebra i s non-superatomic i f and only i f i t has a subalgebra isomorphic to F Q Hence, by Prop, 0.1, any non-superatomic algebra has an epimorph with a dense subalgebra isomorphic to F ; that i s , any non-superatomic q algebra has an atomless, separable epimorph. Hence U(M) = 0, and the rest follows from Theorem 3.5. §4, Some Upper Radicals It i s c l e a r that a c l a s s c o n s i s t i n g of a single atomless, homogeneous algebra i s a hereditary c l a s s , and so generates an upper radical. Any semi-simple algebra, then, can be represented d i r e c t power of t h i s algebra. a l l have t h i s feature. as a sub- The r a d i c a l s defined i n t h i s section In only one case do we make further'mention of t h i s f a c t , f o r i t y i e l d s a subdirect power representation for atomless, complete algebras. 3.7 Definition; E a i s the upper r a d i c a l determined by the i n f i n i t e free algebra F , a 3.8 Proposition; Proof: If a < B, then £ The containment follows e a s i l y from the f a c t that F epimorph of Fg. 3.9 G U Furthermore, F Definition; 3.10 « G Q Proof t a i s i n E^ but not i n E , a a If a $ 8, then G i f and only i f F a and F 8 0 3,. 11 , C o r o l l a r y : Q i s contained i n Gg. Furthermore, are epimorphs of one another, F^ i s an epimorph of Fg. Proof; i s an 6^ i s the upper r a d i c a l determined by F , Proposition; P i s properly contained i n E^. o The rest i s obvious. Let K, « exp.£L .and l e t o a,,(3 .« jy. Then G„ » By the Corollary to Prop. 0,8, any two complete algebras of c a r d i n a l i t y exp H Q For F ft are epimorphs of one another. and F 8 to be mutual epimorphs, i t i s necessary that their c a r d i n a l i t i e s be the same: H a 0 ** °<> We do not know whether the c a r d i n a l i t y condition i s s u f f i c i e n t . 3.12 Theorem; Any atomless, complete algebra i s a sudirect power ofF . D Proof; Using Theorem 3.5, i t s u f f i c e s to show that any atomless, G. p complete algebra i s G^-setai-simple, Any p r i n c i p a l i d e a l of an atom- l e s s , complete algebra i s i t s e l f atomless and complete, and so, by §3, j ) of the P r e l i m i n a r i e s , i t must have F be i n G as an epimorph and Q cannot Hence any atomless, complete algebra i s G -semi-simple. qC 0 We s h a l l see that G Q has many upper r a d i c a l characterizations and so y i e l d s many subdirect product representations f o r atomless, complete algebras. Because of i t s s p e c i a l properties, however, F Q i s an e s p e c i a l l y appropriate b u i l d i n g blook. 3,13 Example: Let Q be the algebra of a l l f i n i t e unions of l e f t - closed, right-open subintervals of the u n i t i n t e r v a l [0, 1) of the reals. I t i s clear that Q i s an atomless, homogeneous algebra. The "Stone space of Q i s the set X obtained from the closed u n i t i n t e r v a l [0, 1] of r e a l s by s p l i t t i n g every i n t e r i o r point x into two parts, — x + and x , — 0 < x We consider X as an ordered set with the natural order: + < x + < y~ < y order topology Q i s i n both G < 1 whenever 0 < x < y < 1, and give i t the T (see S i k o r s k i [24], example §9, E), Q and E , C If Q i s not i n G , Q words, we can embed gN i n X, so t h i s i s impossible, X, We w i l l show that ' then i t has P But Q as an epimorph; i n other |x| «= exp H Q ' and j gN j « exp exp Ho H If Q i s not i n fc , then we can embed the Cantor set 2 in To show that t h i s i s impossible, we show that any uncountable Q closed subspace of X has an uncountable base. q9 Let F be any uncountable closed subspace of X and l e t {G > a be a base f o r F. of the form x countable. val [ x + s Let F + (respectively F~) be the points of F (respectively x~) , + Suppose F Then one of F , F~ must be un+ i s uncountable. + 1] i s clopen i n X and so F O there i s a basic set G then x + x such that x i [ y , 1] and so x + + i G. uncountable. If F + i n F , the i n t e r + [ x , 1] i s clopen i n F. e G^Q F f l [x+, 1], If x + x Thus + Thus f o r d i s t i n c t x , y y we get d i s t i n c t basic open sets G + For any x + and Gy, and the base {G } Q < y , + + in F , + must be i s uncountable, an obvious modification of the argument y i e l d s the same r e s u l t . §5, The Upper Radical Determined bjr Homogeneous Algebras 3.14 Definition: J i s the upper r a d i c a l determined by a l l atomless, homogeneous algebras. J i s contained i n any upper r a d i c a l determined by a c l a s s of atomless, homogeneous algebras. Thus i t i s contained i n both G E . Q Q and It i s not equal to t h e i r intersections however, f o r the algebra Q of the l a s t example i s J-semi-simple, whether J = (?, A natural question here i s In Chapter Seven, we w i l l discuss one consequence of a p o s i t i v e answer. We present some considerations which make a p o s i t i v e answer reasonable. 3.15 Definition; A monotonic c a r d i n a l property v assigns a car- dinal number v(A) to each algebra A i n such a way that v ( A ) $ v(A) x for a l l non-zero x i n A. If t h i s inequality i s an e q u a l i t y f o r a l l non-zero x In A, then A i s c a l l e d v-homogeneous. 3.16 Lemma: If v i s a monotonic cardinal property, then the v-homo- geneous elements of any algebra A are dense i n A. Proof: For any non-zero x i n A, pick y £ x such that v(A^) i s minimal among the v ( A ) f o r 0 $ z < x. By monotonicity of v, A g y i s v- homo- geneous. 3.17 Corollary: If v i s a monotonic c a r d i n a l property, then any complete algebra i s a product of v~homogeneous algebras. Proof: I t i s clear that the property of being v-homogeneous i s her- editary, so we can apply Lemma 2,10, This l a s t r e s u l t i s due to Pierce [22] and provides support f o r h i s conjecture [23] that every complete algebra i s a product of homogeneous algebras. that J * 0 3.18 J v We provide similar support f o r the conjecture e Proposition: Let v be a monotonic cardinal property and l e t be the upper r a d i c a l determined by the class of atomless, v-homo- geneous algebras. Then J v = 0 a Proof % Any non-superatomic algebra has an atomless epimorph, has an atomless, v-homogeneous p r i n c i p a l i d e a l . only superatomic algebras. Hence J v which can contain CHAPTER FOUR CRAMER*S RADICALS We investigate the radicals introduced by Cramer i n [5], §1.The Classes Superatomic Boolean algebras can be characterized as those having no countable f r e e subalgebra. Cramer generalized t h i s as follows: ^•^ Definition: The c l a s s C Q i s the c l a s s of a l l Boolean algebras with no subalgebra isomorphic to F , fl Cramer's proof that the C g i c a l methods. 4.2 a i s a radical class. By p r o j e c t i v i t y , whenever F A, then F a are r a d i c a l classes uses topolo- We present an algebraic proof. Proposition; C Proof: Q can be embedded i n A, a can be embedded i n an epimorph of Thus C Q i s closed under epimorphs. Now suppose that A i s an algebra such that every epimorph of A has a principal ideal i n C , a F a If F a can be embedded i n A, then by Prop. 0.1, can be densely embedded i n some epimorph B of A, If x i s any non- zero element of B, there i s a non-zero y $ x such that y e F . Then the p r i n c i p a l i d e a l of F Q generated by y can be embedded i n which can be embedded as a subalgebra i n B , Since x i s homogeneous, t h i s says that no non-zero p r i n c i p a l i d e a l of B can be i n C , contra- a d i c t i n g the assumption on A. C a Thus A i s i n C , and by Prop, 1,21, a i s a radical class, 4.3 Proposition; If a < 6, then C i s properly contained i n Cg. a Proof: The containment follows from the f a c t that F i n Fg. It i s proper since F § 2 . The C 4.4 a e Cg but F a a a a as an epimorph, Proposition (Cramer [ 5 ] ) : Let Hg 4.6 i C. The class X> i s the class of a l l ,Boolean,,algebras which do not have P Proof: can be embedded as Upper Radicals Definition: 4.5 Q a exp H^, ra Then V A •= Cg. T h i s follows immediately from Prop, 0,6, Corollary: V A Is a r a d i c a l c l a s s , and i f a < 8, then P q is properly contained i n Pg, Except f o r the fact that P of P Q suggests an upper r a d i c a l . If a i s not atomless, the a exp H aS description however, since P and Fg are epimorphs of one another, we immediately get Cg => P A c t u a l l y , we can extend t h i s r e s u l t . Q a •= Gg. 4.6 Proposition; Proof; Since F i s contained i n G has F ffi a a then ft i s semi-simple, |F | «= and C ° G^ i f and only i f Q as a subalgebra, i t i s not i n C homogeneous, i t must be C -serai-siiaple. for which F q9 so C Q But G i s the largest r a d i c a l A i s contained i n G^ If H t and so i t i s an epimorph of F „ a and being a If F a Q In other words, G of the r a d i c a l s . suppose that ^ i s contained i n C , a Q "Hat i s embedded i n Q an algebra A, then by i n j e c t i v i t y of F , we get that A has F epimorph. ° a Q as an and we get e q u a l i t y Conversely, suppose the r a d i c a l s are equal, and > // ° 0 a By c a r d i n a l i t y , then, F a an epimorph and so F a i s in G , But F A a £ C, Q a cannot have F as Q so t h i s c o n t r a d i c t s the equality of the r a d i c a l c l a s s e s . §3« The Radical V Since P Q O and F are epimorphs of one another, V RA N O r r " O G . O We have already seen that a l l atomless, complete algebras are ^ - s e m i simple. This s e c t i o n plays v a r i a t i o n s on the theme that P -algebra8 Q are i n a very strong sense the opposite of complete algebras. The basic fact we need i s Pierce's r e s u l t (53, j ) of Preliminaries) that any i n f i n i t e epimorph of a c-complete algebra must have P morph. Q An immediate consequence of t h i s i s the following: as an epi- 4.7 Proposition; Let A be an i n f i n i t e P - a l g e b r a . Q Then: i ) A has no i n f i n i t e o-complete epimorphs, i i ) A i s not the epimorph of any o-complete algebra, and i i i ) A has no i n f i n i t e , complete subalgebra. 4.8 Proposition: Let V be any c l a s s of atomless algebras s a t i s f y i n g : i ) each algebra i n V has P q as an epimorph, and i i ) there i s an algebra i n V which i s an epimorph of P . Q Then the upper r a d i c a l determined by V i s P , Q Proof; so V Q By condition i ) , a P - a l g e b r a cannot have an epimorph i n Y Q i s contained i n the upper r a d i c a l determined by V, t By con- d i t i o n i i ) , any algebra i n the upper r a d i c a l determined by V cannot "have"P as an epimorph, and so i s *ih* P . 0 4.9 Q Corollary: V Q i s the upper r a d i c a l determined by any of the following c l a s s e s ; i ) a l l atomless, complete algebras, i i ) a l l atomless, a-complete algebras, i i i ) a l l atomless, complete homogeneous algebras, i v ) the c l a s s c o n s i s t i n g of a l l p r i n c i p a l i d e a l s of P / I » A where I i s the i d e a l of f i n i t e sets i n P , a v) the c l a s s c o n s i s t i n g of F , Q vi) the c l a s s c o n s i s t i n g of any atomless, complete algebra of c a r d i n a l i t y exp h{ . CHAPTER FIVE DECOMPOSITIONS OF COMPLETE ALGEBRAS The product decompositions of t h i s chapter depend on f i n d - ing a dense subset of P-elements i n an algebra, f o r some hereditary property P. The search f o r P-elements below an a r b i t r a r y non-zero element leads naturally to the consideration of various c h a i n - l i k e conditions. §1. The General Setting We have already seen that Pierce's decomposition of complete algebras v i a c a r d i n a l properties i s a s p e c i a l case of Lemma 2.10. theorems of this chapter also make use of t h i s lemma. The We note that the properties of being homogeneous and unequivocal are hereditary properties. 5.1 Hence, we immediately get the following: Proposition: Let A be a complete algebra. Then A i s a product of homogeneous (unequivocal) algebras i f and only i f the homogeneous (unequivocal) elements of A are dense i n A. Pierce's r e s u l t also included a uniqueness feature (see[22J) which also holds f o r decompositions into homogeneous (unequivocal) algebras, whenever such decompositions exist. The following propo- s i t i o n includes a l l these uniqueness r e s u l t s as s p e c i a l cases. 5.2 Proposition: Let P be a hereditary property, and suppose A i s a complete algebra with a dense subset of P-elements, Suppose there i s an equivalence r e l a t i o n = on the P-elements of A such that: (*) i f x and y are P-elements of A and x ji y, then x A y = 0. Then there i s a unique decomposition A - H(A : x e X) with the f o l l o w - ing p r o p e r t i e s : i ) f o r any x e X, A x i s a product of P-algebras A , y where y-^ = yg f o r any y^, y2 e Y^p i i ) for x i Proof: z i n X, y E Y , X and e Y^, y 2 e Y t y^ t y* 2 The set X consists of the suprema of the equivalence classes of P-elements of A, Then X i s d i s j o i n t e d by (*), and sup X = 1 by the density of P-elements i n A. The rest of the proof i s a s t r a i g h t - f o r - ward v e r i f i c a t i o n . To see how t h i s a p p l i e s , v/e need to specify an equivalence r e l a t i o n f o r each of the properties we have considered: i ) v-homogeneity: say x = y i f ^(A^) » v(Ay), i i ) homogeneity: say x = y i f A^ - A , y and i i i ) unequivocality: say x = y i f L(A ) = 1 ( A ) . y In each case, i t i s easy to see that condition (*) i s s a t i s f i e d , so x that a suitable replacement of P i n Prop, 5,2 w i l l y i e l d a uniqueness r e s u l t i n each of these three situations, §2, Decompositions into Homogeneous Algebras 5.3 Definition; An algebra A w i l l be c a l l e d near-homogeneous i f every descending chain of p r i n c i p a l ideals of A contains only a f i n i t e number of isomorphism types of algebras. F i n i t e products of homogeneous algebras are a natural example of near-homogeneous algebras. set algebras. Another class of examples are the power- Any p r i n c i p a l i d e a l of such an algebra i s another power- set algebra, which i s determined up to isomorphism by the c a r d i n a l i t y 6f i t s atoms. Hence any descending chain of such ideals y i e l d s a descending chain of c a r d i n a l s 5.4 B which must be f i n i t e . Theorem; If A i s complete and near-homogeneous, then A i s a pro- duct of homogeneous algebras. Proof: By Prop. 5.1, i t s u f f i c e s to show that any non-zero p r i n c i p a l i d e a l of A contains a non-zero homogeneous p r i n c i p a l i d e a l . x be a non-zero element of A, If A non-zero element y 5 x such that A So l e t i s not homogeneous, there i s a i s not isomorphic to A . Proceedy x ing i n d u c t i v e l y , we get a descending chain A >A >,.,> A > ... x y z where no two adjacent algebras are isomorphic. By §3, g) of the P r e l i m i n a r i e s , i f any two algebras i n the chain are Isomorphic, then they are also isomorphic to a l l the intervening ones. no two algebras i n the chain can be isomorphic. Hence Because A i s near- homogeneous, the chain must terminate i n a f i n i t e number of steps, and the algebra thus obtained i s a non-zero homogeneous p r i n c i p a l i d e a l of A contained i n A,, x 5,5 Example; We present an example of a complete algebra A which i s a product of homogeneous algebras, but which i s not near-homogen=» eous. Let i satisfying K exp // and K o K n ° " ^n* n n s n < w} be a s t r i c t l y increasing sequence of cardinals + 1 = exp Such sequences e x i s t : f o r example, take K = Q For any such sequence, l e t A denote the n normal completion of the free algebra on tc generators. Then A n Is Q complete, homogeneous, and i t s c a r d i n a l i t y i s tc^. Let A *» H(A : n < u). n There i s a n a t u r a l isomorphism between c e r t a i n p r i n c i p a l i d e a l s of A and p a r t i a l products of the A^ as follows: B^ Thus we get a descending chain of ideals of A: B Q a E(A : k « n < w). n > B^ > ... > B^ > Any non-zero p r i n c i p a l i d e a l of B^ must contain a copy of some A f o r n £ i and so must have c a r d i n a l i t y at l e a s t B^ Q For j < i , howevery, Bj has a p r i n c i p a l i d e a l isomorphic to Aj of c a r d i n a l i t y Kj. Since Kj < K^, B^ cannot be isomorphic to Bj. §3. Decompo s i t ion s into Unequivocal 5.6 Definition: Algebras Let {K > be a well-ordered chain of r a d i c a l classes. a We say that an algebra A i s R-layered either A A i f , f o r any non-zero x i n A, i s f i n i t e , or there i s an ordinal 0 such that L(A ) « R . X P 0 A w i l l be c a l l e d layered i f there e x i s t s some well-ordered chain of r a d i c a l classes {K > such that A i s R-layered, a Examples of layered algebras w i l l be given i n §4. I t i s clear that any p r i n c i p a l ideal of an R-layered algebra i s i t s e l f Rlayered, 5.7 Theorem: If A i s a complete layered algebra, then A i s a pro- duct of unequivocal Proof: algebras. By Prop. 5.1, i t s u f f i c e s to show that any non-zero p r i n c i p a l i d e a l of A contains a non-zero unequivocal p r i n c i p a l i d e a l . So l e t x be a non-zero element of A, and suppose {& ) i s the chain of r a d i c a l s a with respect to which A i s layered. If contains an atom, then i t contains the unequivocal algebra 2. Otherwise, i f A i s atomless, l e t & be the l e a s t a such that I (Ay) «• R some y f o r which l(Ay) » R^, Rg, q f o r some non-zero y .$ x. Choose Now for 0 ^ z •$ y, i ( A ) i s contained i n z Since A i s R-layered, 1(A ) i s some R , and by the minimality of Z &, L ( A ) <= Rg, Z ideal of A^. But then, by Prop, 2,15, A a i s an unequivocal principal 5.8 Theorem; Let A be a complete algebra, and suppose that f o r any non-zero x i n A, there i s a power-set algebra P such that A„ and P x are epimorphs of one another. Then A i s a product of unequivocal algebras. Proof; Using Theorem 2.39, i t i s clear that the condition on A i s p r e c i s e l y what i s needed to make A a P-layered algebra f o r the chain Using Theorem 2,39, we see that i s would be extremely u s e f u l , i n determining the scope of t h i s theorem, to know which algebras, other than power-set algebras and completions of f r e e algebras, have large f r e e subalgebras. Unfortunately, l i t t l e i s known. As a sample, we quote the f o l l o w i n g . r e s u l t : 5.9 Proposition (Efimov [11]); For any algebra A, l e t cA denote the supremum of the c a r d i n a l i t i e s of f a m i l i e s of d i s j o i n t elements of A. Suppose A i s an algebra such that cA 4 K and |A| > exp exp exp <c. Then A has a free subalgebra on (exp tc)+ generators. We note that F Q can never be i n the lower r a d i c a l by an i n f i n i t e complete algebra. generated Hence there i s no point i n attempting further r e s u l t s along these l i n e s using the chains {F }f Q o r ' §4, Connections with Cardinal Properties It i s possible to obtain Theorem 5,7 i n a s l i g h t l y more lengthy manner using Pierce's r e s u l t (Cor, 3.17). There are some i n t e r e s t i n g a d d i t i o n a l r e s u l t s along the way, and the approach Is better suited to presenting examples, so we proceed to develop i t now, 5.10 Definition; For any well-ordered chain of r a d i c a l c l a s s e s {R }, we say an algebra A i s admissible with respect to the chain i f a i t i s i n one of the r a d i c a l s of the chain. For any admissible A, we define p(A) = min {H i A e R }. a Then p i s a c a r d i n a l property on admissible algebras, and the f a c t that i t i s monotonic follows e a s i l y from the fact that every r a d i c a l c l a s s i s hereditary. 5.11 Lemma; We note that p(A) i s always i n f i n i t e . The admissible algebra A i s p-homogeneous i f and only i f there i s an o r d i n a l 6 such that: i) A e & a f o r a l l a ^ g, i i ) A i s R -semi-simple f o r a l l a < 6. a In t h i s case, of course, p(A) = H^Proof: Let A be p-horaogeneous with p (A) » ft b A i s in R . o e Then f o r a l l a J. 6, By p-homogeneity, p ( A ) = p(A) = ft f o r a l l non-zero x x e i n A. any R In other words, no non-zero p r i n c i p a l i d e a l of A occurs i n a f o r a < £. Hence A i s R -semi-simple f o r a l l a < 6. a v e r s e l y , suppose such a g e x i s t s . i s R -semi-simple Ct f o r a < 8, A X Since p ( A ) zero x i n A. x Then c l e a r l y p(A) « H^, cannot be i n any such R Cl ConSince A f o r any n O n - p (A) •=• K g , we must have p (A^) " /fg« Hence A i s p-homogeneous. 5.12 Lemma; Let {R } be a well-ordered chain of r a d i c a l s . Q the algebra A i s R-layered Suppose (hence admissible) and p-homogeneous. Then A i s unequivocal. Proof; 11 1 Let p (A) =» // • and l e t x be a non-zero element of A. ot) By P p-homogeneity, A cannot be i n R f o r any a < 8. x " Hence, f o r any such q, • Lj(A •) f .(?«'. Since A i s R-layered and_i-,(A ) i s contained i n Rg, we a must have L (A ) => R „ f o r any non-zero x i n A, x p Hence A i s unequivocal, We are now ready to re-prove Theorem 5.7: 5»13 Theorem: If A i s a complete layered algebra, then A i s a pro- duct of unequivocal algebras. Proof: Suppose A i s R-layered. p-homogeneous algebras. By Cor. 3.17, A i s a product of Being p r i n c i p a l ideals of A, these algebras are also R-layered and so each i s , i n f a c t , unequivocal. We are now ready to proceed with examples. We concentrate on the chains {P } and {P > which define c a r d i n a l properties u and 6 a a respectively. Note that every complete algebra i s admissible with respect to these chains. GCli. For the remainder of t h i s chapter, we assume Aside from the fact that GCH s i m p l i f i e s the examples we consider, the following proposition requires the assumption that i f a < 6, then exp H a 5.14 < exp # . p Proposition (GCH): Furthermore, 6(A) Proof: Since P a s For any admissible algebra A, 6 (A) < ir(A)+. ir(A)+ i f and only i f L(A) = P^ where # cannot have P i * as a + This implies the f i r s t statement. a n epimorph, we get P 5.15 Q » if (A). C. # ^» A C l e a r l y , <S (A) «• it(A)+ i f and only i f A t P„ but A £ V ; that i s , A e P„ and A has P Theorem 2.39, a as an epimorph. t h i s i s equivalent to L(A) » P , C o r o l l a r y (GCH): An algebra A i s P-layered i f and only i f , for every non-zero x i n A such that A i s i n f i n i t e , 6(A ) = ir(A )+, X . x X A complete algebra with t h i s property i s a product of unequivocal algebras. 5.16 Examples (GCH): We note that ir(P ) = H a a and 6(P ) =» %ot+l« h' » suppose a Q Hence every power-set algebra i s P-layered. Since P^ *° L ( F ^ ) , we have * ( F a + i s a limit ordinal. so F so a E P, We always have that'-F Suppose B < a and F a } ft °, a A + 1 a e Pg ) 83 a Now i s an epimorph of Then F Q P, a i s an epimorph of Pg But 6 < o implies B + 1 < a and K , V i + .< H a °» By So ¥ i s not i n Pg, 0 < a, and ir(F ) » a 0 a . In e i t h e r case, T K F ) Q ° £(K : y < a ) . y Since a + i P , so does F + i . Q F a+1 s P~ l a y e r e d a as an epimorph, then, by i n j e c t i v i t y of Hence 6 (^ 0 i has P F a+A ^ "^a+l*' ^ o r s u c c e s s o » For l i m i t o r d i n a l s , the s i t u a t i o n i s unclear. we do not know i f F i s P-layered. H 8 „ » then IF :S r morph; that i s , F an epimorph. a e V. a — I " H„ « f(F„), cannot have P as an e p i - If 6 < a, 'then 3 + 1 < o and so F Q n so F t So then does F , and F then, <5(F ) = H For example, However, i f a i s a l i m i t o r d i n a l — satisfying H , ° ordinals, r a t f o r 8 < a. Q has Pg as In t h i s case, Hence, i n t h i s case, F„ i s not P-layered. We note that such ordinals e x i s t ; f o r example, take a to be the f i r s t „.uncountable„o.rdinal« CHAPTER SIX CLOSURE PROPERTIES OF RADICAL AND SEMI-SIMPLE CLASSES Every r a d i c a l class i s closed under f i n i t e products and weak products. products. Every semi-simple class i s closed under subdirect We extend these r e s u l t s . §1. Closure of Radical Classes under Products Any K -product of algebras has P a as an epimorph,, so t h i s gives us a crude negative r e s u l t : whenever a r a d i c a l does not contain P a o then n o - p r o d u c t can be i n the r a d i c a l . the converse: whenever P under ^ - p r o d u c t s . One might hope to show i s i n a r a d i c a l c l a s s , then i t i s closed Cramer [5] has obtained a r e s u l t which shows that this i s false. 6.1 P r o p o s i t i o n : For any ordinal a, there i s a sequence i^ ' n < w} n of superatomic algebras whose product i s not i n C « a We s h a l l see In Chapter Seven that any ? a i s contained, i n some Cg, so we can f i n d a sequence of superatomic algebras whose product i s not i n P , Q In Chapter Seven, we w i l l present a weaker form of t h i s (false) conjecture which has more l i k e l i h o o d of being true. 6.2 Proposition: If exp ft a ra exp ft then P^ i s closed under ft t products of complete P -algebras. a Proof; Let A = Il(A^: i e I) be a product of complete P -algebras with Q | l | « H$« By Theorem 2.37, |A | « exp ft and so |A| <S (exp # / ^ ± a 0 exp ft » Since t h i s product i s complete, i t i s i n ? a a - by Theorem 2,37. Halmos ([14], exercise 3, p, 118) has defined a "weak product" s l i g h t l y d i f f e r e n t from our weak product. We give a generaliza- tion of h i s construction, which also includes our weak product and the construction of Example 2.32 as special cases. For any product A of algebras, P(I) i a embedded i n A <= Jl(A^: i e I) as the elements of A with {0, l}-coordinates. Let B be a subalgebra of P(I) which contains *th£"atoins' of A'P (1). , 6.3 Definition; The product of the A^ over B_ i s the subalgebra C of A consisting of a l l elements which d i f f e r from an element of B i n at most a f i n i t e number of coordinates. It i s clear that C i s the subalgebra of A generated by B and wB(A^; i e l ) . Halmos' "weak product" corresponds to choosing B = P(I)« For the weak product, choose B to be the f i n i t e - c o f i n i t e algebra on I, Our Example 2,32 used a superatomic subalgebra of P ( I ) . 6.4 Lemma: Let J be the i d e a l of C generated by the A^ and l e t K be the ideal of B generated by i t s atoms. Then C/J - B/K. Proof: I t i s c l e a r that C contains the that x d i f f e r s from elements b^, coordinates. Let x e C and suppose e B i n a t most a f i n i t e number of Then b-^ d i f f e r s from b2 i n at most a f i n i t e number of coordinates; that i s , b^ +. b£ e K. Thus the map which sends x to the coset of bjL i n B/K i s a well-defined epimorphism. Clearly, J i s i t s kernel. 6.5 Let R be a r a d i c a l c l a s s and suppose the A^ and B Proposition: are i n R . Proof: Then the product C of the A^ over B i s i n R . J i s generated by r a d i c a l i d e a l s of C, and C/J, being isomor- phic to an epimorph of a r a d i c a l algebra, i s r a d i c a l . 6.6 Corollary: Thus C e R . Radical classes are closed under weak products. 52. Closure of Radical Classes under Coproducts 6»7 Easic Lemma: Let R be a r a d i c a l c l a s s . Suppose A and B are Boolean algebras, and that A i s the summit of a ladder {1 } with the Q following property: (*) for each a, each element of I j _ / I a + is a finite j o i n of cosets [aJ such that (A/I ) ^ j + B e R . a Then A + B e R . Proof: Let K Q be the ideal of A + B generated by I . Q Then {K^} i s a ladder i n A + B with summit A •+. B, and so, by Lemma 2.3, i t s u f f i c e s to show that K a + A /^ e ^ £ ° a l l <*, ^a+i/^a r a A + B/K^, which, by Prop e i s a Ideal °f n 0,4, i s isomorphic to A / I + B a 0 As an ideal of A / I + B„ the elements of K ^ ^ / i ^ can be represented as f i n i t e 0 joins of elements of the form [a] A b where [a] e I By (*)„ [a] » [ a ] v ... v [ a ] where ( / A x n I 0 )[ a i a + i /l a ] + B e R, and b e B, This l a t t e r algebra i s isomorphic to (A + B/K, ) , which contains (A + B/K ) . , , laiJ . a [a^jAb r r a Then, since each [a^l A b i s a r a d i c a l element of Kg^/K » we get that K j / K Q + 6.8 e R as required, a Theorem: Let R be a r a d i c a l c l a s s and l e t B e R. Then, f o r any superatomic algebra A, A + B e R, Proof: By Prop, 2,24, A i s the summit of a ladder { l } where every o element of I + i / I a a i s a f i n i t e j o i n of atoms [ p i e A/I^. ( A / l ) j + B « = 2 + B= a 6.9 [ p Then BeR. Corollary (Day [ 7 ] ) : A f i n i t e coproduct of superatomic algebras i s superatomic. 6.9 Example: Let Z be the Boolean space (under the order topology) of ordinals l e s s than or equal to ft, the f i r s t uncountable o r d i n a l . Then the algebra S of clopen subsets of Z i s superatomic, and so, by the l a s t theorem, A >• F-. + S e F , We show that i f I i s the i d e a l of o o A generated by the elements x such that A^ - F , then A/I Is not Q superatomic. Let Y.» S(F ) be the Cantor set. Then X - Y x Z i s the Stone space of A. Let U » S ( I ) . For any clopen subset M of X, Mfl U must have a countable base Now suppose M i s a clopen subset of X B such that (y, &) e M f o r some y £ Y, The p r o j e c t i o n P^[Mj of M onto Z i s an open subset of Z containing Q. This open set contains a clopen set N which i s homeomorphic to Z, whose pre-image p ~*[Nj i s a clopen z subset of X contained i n M. then, can M. Since Z has no countable base, neither, Hence U n {(y, 0): y e Y} = <j>. C l e a r l y , { (y, Q): y e Y} i s homeomorphic to Y, so X - U has a closed subspace homeomorphic to Y. A l g e b r a i c a l l y , A/I has F 6.10 Lemma: morphs. as an epimorph, and so cannot be superatomic. Let H be a c l a s s of Boolean algebras closed under e p i - Suppose B i s an algebra whose coproduct with any f/-algebra i s i n JL(tf). Proof; q Then the coproduct of B with any L(H)-algebra i s i n L(H). Let A e L(tf). Then i t i s the summit of i t s tf-ladder { I } . a and every element of I + i / I a a i s a f i n i t e j o i n of H-elements. By the assumption on B, i t s coproduct with any p r i n c i p a l i d e a l generated by an H-eletsent must be i n L(tf). 6.11 Theorem: coproducts. Proof: Hence, by Lemma 6.7, A + B e L (H) . Let X be any c l a s s of algebras closed under f i n i t e Then L(X) i s closed under f i n i t e coproducts. Let H be the class of a l l epimorphs of X-algebras. Since the coproduct of epimorphs of two X-algebras i s an epimorph of t h e i r coproduct, and since X i s closed under f i n i t e coproducts, so then i s H. If C i s an H-algebra, then i t s coproduct with any other H-algebra i s in H and so i n L(H). By Lemma 6,10, the coproduct of C with any L(H)- algebra B i s i n L(H). Since t h i s i s true f o r any C i n H, we have that the coproduct of B with any H-algebra i s i n L(H). Applying Lemma 6.10 again, we get that the coproduct of B with any L(H)-algebra L(H). LOO, Hence L(H) i s closed under f i n i t e coproducts. Since L (X) = the r e s u l t follows, 6.12" Corollary; Proof: A i s in i s closed under f i n i t e coproducts. The c l a s s of algebras of c a r d i n a l i t y at most H a i s closed under f i n i t e ( i n f a c t , ^ ~ ) coproducts. a One might hope to prove that f i s closed under H -coproducts. ct Vie w i l l show, i n the next section, that i t i s not even closed under countable coproducts of superatomic algebra. The following r e s u l t i s from Cramer [5]: 6.13 Proposition: Suppose that {A : i e 1} i s a c o l l e c t i o n of C ~ i Q algebras, f i n i t e coproducts of which are i n C . £fg and that H a the sum of H c ; Suppose that |l| » i s //^-inaccessible (that i s , ft cannot be expressed as a cardinals each of which i s l e s s than //„). Then the coproduct A •» Z (A.: i e I) i s i n C . . l a Proof: The coproduct A i s the union of a l l i t s subalgebras B^, j e J , which are f i n i t e coproducts of the A^ (see Preliminaries, §4, c ) . that | j | « |X| « H . Note If A has a free subalgebra generated by a set D of c a r d i n a l i t y H i then since i s K^-inaccessible, D H a c a r d i n a l i t y hf a f o r some j e J . must have But D n Bj generates a f r e e subalgebra of Bj, contradicting B^ e C^. 6.14 Corollary: Suppose that Let { A ^ i e 1} be a c o l l e c t i o n of F -algebras. a | l | = H$ and that H i a+ i s /^-inaccessible. Then the co- product of the A^ i s i n Proof: Since F^ does not have F ^ as a subalgebra, F + a i s contained in C ^ , and F^ i s closed under f i n i t e coproducts. a + I n t u i t i v e l y , t h i s says that small enough coproducts of small enough algebras cannot have large free subalgebras. *In-viev7vof >,fche f act that f i n i t e products ,of,projective a l gebras are p r o j e c t i v e , i t i s not unreasonable to ask i f f i n i t e ucts of complete algebras are complete. coprod- It i s relevant i n t h i s con- text since a p o s i t i v e answer would have consequences concerning the closure of P and possibly the ? a under f i n i t e coproducts. Unfortunate- l y , the answer i s almost always negative. 6.15 not Proposition: If A and B are i n f i n i t e algebras, then A + B i s complete. Proof: Choose i n f i n i t e d i s j o i n t c o l l e c t i o n s {a^: i < u} and {b^: i < «} i n A and B respectively. Set x^ ° a^A b^ i n A + B. Let x e A + B be an upper bound of {x^: i < u} . We show that x cannot be a l e a s t upper bound. form x = (c^ A d-^) v We note that x can be represented .,. v ( c A d ) where c^ e A and n x j A (c^ A d^) ^ 0 S e B. n there e x i s t k and i £ j such that x^ A (c^ A i n the Then d^) $ 0 and f o r otherwise, x would i n t e r s e c t only finitely In other words, (a^ A CJ^) A (b^ A d^) ^ 0 and many x^. (a^ A c^) A (bj A d^) f 0. Then, since A and B are independent algebras of A + B, we get that y » (a^ A C^) A since i ^ j , either a.^ A a any m < w, XJJJ A y <° 0 f o r a l l m < u. m (bj A d ) k = 0 or bj A b m & 0. » 0. subFor Hence But 0 £ y •$ x so x A y' i s an upper bound of ( x : n < wl which i s s t r i c t l y smaller than x. n This proposition generalizes Exercise 6N of Gillman J e r i s o n [12], where i t i s asserted that P 6.16 Corollary: only i f one Proof: a and + P^ i s not complete. The coproduct of two algebras i s complete i f and i s f i n i t e and the other complete. This follows immediately from the proposition and that i f A i s f i n i t e with n atoms, then A + B - the f a c t B. n Using the f a c t that 0 i s closed under f i n i t e coproducts, we are able to prove the following: 6.17 Proof: Proposition: E Q i s closed under f i n i t e coproducts. Suppose that A and B do not have an epimorphisra of A + B onto F as an epimorph. with kernel K, l e t I » K f) A If f i s and J » K n B be the corresponding i d e a l s i n A and B. i d e a l of A + B generated by I and J . A + B/L has A + B/K - F Q Since L i s contained i n K, as an epimorph. A onto the subalgebra A/I of F « o The eplmorphism f maps Hence A/I i s countable, so the only atomless epimorph i t could have i s F . Q occur, so A/I i s superatomic. Let L be the Because A e E , t h i s cannot Q S i m i l a r l y , B/J i s superatomic, and so, then, i s t h e i r coproduct A/I + B/J. However, A/I + B/J - A + B/L, and we have already shown that t h i s algebra has F Q as an epimorph. T h i s contradiction shows that A + B .e E . o 6.18 X. n Corollary; Let X be the product of the Boolean spaces X^, a ... Then X has a subspace homeomorphic to the Cantor set 2 ° i f and only If one of the does. §3. Coproducts and Semi-Simplicity The r e s u l t s of t h i s section indicate that coproducts are f a r more l i k e l y to be semi-simple than r a d i c a l . 6.19 Theorem; Let R be a r a d i c a l c l a s s and suppose the c o l l e c t i o n {A^: i e 1} contains at l e a s t one R-semi-simple algebra. Then the coproduct of the A^ i s R-semi-simple. Proof; C l e a r l y , i t s u f f i c e s to show that i f A i s R-semi-simple, then so i s A + B f o r any B, By Prop, 1.25, we must show that any non-zero p r i n c i p a l i d e a l of A + B has an R-semi-simple epimorph, and i t i s c l e a r that we can r e s t r i c t our a t t e n t i o n to non-zero elements of the form a A b where a e A and b e B. But (A + B) . aftb c A a + B, and A , b a* which i s R-semi-simple, i s a r e t r a c t of t h i s coproduct. Kence A + B i s R-semi-simple. Any i n f i n i t e coproduct of algebras i s atomless; that i s , 0-semi-simple. We are able to obtain analogous r e s u l t s f o r any r a d i c a l , provided we r e s t r i c t ourselves to the coproduct of i n f i n i t e l y many copies of the same algebra, 6.20 Definition; Let K be any cardinal and A any algebra. We write KA f o r the coproduct of K copies of A and c a l l i t a tc-multiple of A, 6.21 Proposition; If A i s any algebra and tc any i n f i n i t e c a r d i n a l , then KA i s unequivocal. Proof; Let X •= S ( A ) and suppose M i s a clopen subset of X Then M •» H(M : a a <K) where M Q a l l but a f i n i t e number of a. i s a clopen subset of X and M a S(KA). •= X f o r But then the p a r t i a l product I I ( ^ ; M ft i s a r e t r a c t of M and i s homeomorphic to X , K = A l g e b r a i c a l l y , any non- zero p r i n c i p a l i d e a l of KA has a r e t r a c t isomorphic to KA. Thus the lower r a d i c a l generated by any such p r i n c i p a l i d e a l i s the same as the lower r a d i c a l generated by KA. 6.22 Corollary: In other words, KA i s unequivocal. If A i s not i n a r a d i c a l class R , simple f o r any i n f i n i t e tc. then KA i s R-semi- Proof; Since A has a non-radical epimorph A, i t cannot be i n R. Being unequivocal, i t i s R-semi-simple. 6.23 Corollary; that H S i s F -semi-simple. Q Q Proof; ^a+1' For any .a, there i s a superatomic algebra S such Let S be the f i n i t e - c o f i n i t e algebra on a set of c a r d i n a l i t y s i n c e every non-zero p r i n c i p a l i d e a l of # S Q Ma+l i t must be F -semi-simple. 6.24 Corollary; t has c a r d i n a l i t y a Every algebra i s a r e t r a c t of an unequivocal algebra. The l a s t c o r o l l a r y i s a generalization of the f a c t that every algebra i s an epimorph of a free algebra. Gratzer {13] has -•announced a stronger -result :~for -any algebra 'A, there is'an* algebra B such that A + B i s homogeneous. Thus any algebra i s a retract of a homogeneous one. 6.25 Definition: A r a d i c a l class R i s proper i f i t does not contain a l l Boolean algebras. Then i t cannot contain a l l free algebras. any proper r a d i c a l class R, l e t o(R) <F H a B such that F 6.26 p e Theorem: £ R. where a i s the l e a s t o r d i n a l Note that o(R) i s always i n f i n i t e , Let R be a proper r a d i c a l c l a s s and l e t A be an algebra with more than two elements. Proof: For Let ic = tf > o(R). Then KA i s R-semi-simple f o r a l l K > o(R), By §6, d) of the P r e l i m i n a r i e s , KA has F 0 as a r e t r a c t . By d e f i n i t i o n of a(R), F a t R, so KA £ R. KA i s R-semi-simple. 6.27 Examples: We l i s t c(R) f o r known R: i ) a{0) - H Qt i i ) o(P ) - j j o o i i i i ) o(P) «fc , 0 i v ) o(E ) - f l . a a v) o ( F ) « ^ a a + 1 , v i ) o<C )'-Jf , o a v i i ) o(6 ) - ftg where tf a p v i i i ) o(J) - f y , , ix) o(K) - H v fy* , 0 By Cor. 6.22, CHAPTER SEVEN THE LATTICE OF RADICALS We can p a r t i a l l y order Boolean algebra r a d i c a l s by the r e l a t i o n of containment. If we extend the term " l a t t i c e " to include structures defined on classes as well as sets, we f i n d that the Boolean algebra r a d i c a l s form a l a t t i c e with some i n t e r e s t i n g algebr a i c properties, §1, Lattice-Theoretic P r e l i m i n a r i e s 7.1 ; D e f i n i t i o n : An abstract algebra ^L; v , A , *, 0, pseudo-complemented d i s t r i b u t i v e l a t t i c e i s called a (with 0 and 1) i f <(L; v , A , 0, ]) i s a d i s t r i b u t i v e l a t t i c e (with 0 and 1) and * i s a unary operation on L s a t i s f y i n g a A b «* 0 i f and only i f b $ a*. Thus a* i s the maximum of the elements d i s j o i n t from a, A more general concept i s the f o l l o w i n g : 7.2 Definition: An abstract algebra ^ L ; v , A , 0, l ) i s a Brouwerian lattice i f i t is a lattice (with 0 and 1) i n which, f o r any a, b e L, there i s c e L such that a A x < b i f and only i f x $ c, the element c by (b:a), We denote Setting a* «= (0:a), we see that any Brouwerian l a t t i c e i s pseudo-complemented, and i t can be shown (see Birkhoff [4]) that any Brouwerian l a t t i c e i s d i s t r i b u t i v e . In a complete l a t t i c e , an obvious candidate f o r (b:a) i s the supremum of a l l x such that a A x 4 b. If the l a t t i c e also s a t i s f i e s the i n f i n i t e d i s t r i b u t i v e law: a A sup {a^: i e 1} « sup l a A a^: i e I}, then t h i s w i l l s u f f i c e to show that the supremum c i n questions does indeed s a t i s f y a A 'c < b, and the l a t t i c e w i l l be Brouwerian, 7,3 Proposition (see Lakser [18]): Let ^L; v , A . *» 0, 1^ be a pseudo-complemented d i s t r i b u t i v e l a t t i c e . a) Then f o r any a, b e L: i ) a .$ a**, i i ) a $ b implies b* < a*, i i i ) a* -• a***, i v ) a o a** i f and only i f a ° b* f o r some b e L, v) a = a**, b « b** implies a A b » (a A b ) * * , v i ) 0* - 1, 1* - 0, 0 = 0**, 1 - 1**. b) of L» Let L* «• {a*: a e L} » {a £ L: a =» a**}, c a l l e d the skeleton Then 0, 1 e L* and L* i s closed under A and *, If we define a u b « (a* A b*)* « (a v b ) * , then L* i s closed under u, and ^L*; u,A ,.*, 0, 1^ i s a Boolean algebra,, which i s complete i f L i s . c) of L. Let D = {a e L: a* = 0), c a l l e d the set of dense elements Then D i s a f i l t e r i n L; that i s , i t i s closed under A and l a r g e r elements. 7.4 Lemma; l a t t i c e L. Let p be an atom i n a pseudo-complemented distributive Then: i ) f o r any a e L, p ^ a V a*, and ii) p Proof: 0. a i f and only i f p 4 a**. If p i s not contained i n a, then since i t i s an atom, p A a = Then p < a* and i ) follows. a, then again we get p If p a* so that p 4 a A a** «* 0, c o n t r a d i c t i n g the f a c t that atoms are non-zero. 7,5 Proposition: a** but p i s not contained i n The other d i r e c t i o n of i i ) i s obvious, Let L be a complete pseudo-complemented distribu- t i v e l a t t i c e , and l e t t be the supremum of the atoms of L (we assume there are some). Let L £ « {a e L: a .$ t} B {aAt:aeL}« For any .a . cL, „def ine..a° = a* A ,t» .Then: i ) \ L ; V p A , °, 0, y fc i s a Boolean algebra; i n f a c t , i t i s a power-set algebra, i i ) L^ i s an epimorph of L*, and the epimorphism i s an isomorphism i f and only i f t e D« Proof: Note f i r s t that by Lemma 7.4, t $ a v a* f o r any a e L, Thus t = (a v a*> A t =• (a A t ) v (a* A t) = (a A t ) v a° f o r any a e L. If a e L , then a «= a A t, so we get t » a V a° f o r a l l a e L^., fc l y , a A a° » 0, so L^. i s a Boolean algebra. atomic, i t i s a power-set algebra. Then f c l e a r l y preserves A, Clear- Since i t i s complete and Define f : L * — L f c by f (a) « a A t . Also, f (a*) » a* A t » a°. By Lemma 7,4, a A p ** a** A p f o r a l l a e L. We extend t h i s to t by showing that a A t » sup {a A p: p i s an atom). I t i s clear that a A t i s an upper bound f o r t h i s set. Suppose c i s any other upper bound. atom p, p £(a v a*) A (p v a*) = (a A p) V a* < c V a*. t $ c v a* and a A t « a A ( c v a*) « a A c ^ c. f a c t , the l e a s t upper bound of the set. a A t •» a** A t f o r a l l a e L. fc But then Hence a A t i s , i n I t e a s i l y follows that Thus f (a u b) «= (a v b)** A t « (a v b) A t « (a A t) v (b A t) *» f (a) v f (b). a A t E L , f (a**) = a A t . Then f o r any Furthermore, f o r any Hence f i s an epimorphism. Clearly, a i s i n the kernel of f i f and only i f a A t » 0 i f and only i f a ^ t * . Thus f i s an isomorphism i f and only i f t* » 0; that i s , t e D, 7.6 Proposition; Let L be as i n Prop, 7.5. Then L i s atomic i f and only i f t e D.' Proof; Suppose t e D and that a i s a non-zero element of L containing no atoms. Then a A p » 0 f o r a l l atoms p, so that a A t <= 0, then a $ t * = 0, But Conversely, i f L i s atomic and t t D, then there i s an atom p contained i n t * . But then p < t A t * =» 0. Recalling that f o r any a e L, t «* (a A t ) v/ (a* A t ) , that i f a e D, then t ^ a. we see Hence t i s a lower bound f o r D, and i n case t e D, then D i s the p r i n c i p a l f i l t e r generated by t . In t h i s case, L s p l i t s at t into a p r i n c i p a l f i l t e r above t and a power-set algebra below i t . §2, The L a t t i c e of Radicals f o r Associative Rings Snider [25, 26], using results of Leavitt [19], gave the f i r s t account of the l a t t i c e of r a d i c a l f o r associative rings. The class of such r a d i c a l s forms a complete l a t t i c e under the natural ordering. The meet of any c o l l e c t i o n of r a d i c a l s i s t h e i r i n t e r s e c - t i o n and the j o i n i s the lower r a d i c a l generated by their union. The j o i n i s also determined by i t s semi-simple c l a s s , which i s the intersect i o n of the semi-simple classes of the radicals i n the c o l l e c t i o n . The c l a s s of hereditary r a d i c a l s forms a complete sublattice of the l a t t i c e of r a d i c a l s and i s shown to s a t i s f y an i n f i n i t e d i s t r i b u t i v e law which makes the remarks following Defn. 7.2 pertinent. clude that the l a t t i c e of hereditary t i v e , and pseudo-complemented. We con- r a d i c a l s i s Brouwerian, d i s t r i b u - Snider shows that t h i s l a t t i c e i s atomic, the atoms being the lower r a d i c a l s generated by a s i n g l e simple ring. Hence, using Prop, 7.5 and Prop. 7.6, we can extend h i s r e s u l t s as follows: 7.7 Proposition: Let T be the lower r a d i c a l of associative rings gen- erated by the class of simple r i n g s . i ) T i s hereditary Then: and T* =• 0, i i ) the class of hereditary r a d i c a l s contained i n T form a power-set Boolean algebra under the natural order, and i i i ) t h i s algebra i s isomorphic to the skeleton of the l a t t i c e of hereditary radicals. Snider characterizes (S:R), but there i s an i n t u i t i v e l y more obvious candidate f o r i t than he gives. not, i n general, hereditary. i z a t i o n of (S:R) Unfortunately, I t w i l l , however, y i e l d a nice it is character- i n universal classes f o r which every r a d i c a l i s hereditary. 7.8 Proposition; Let R and S be hereditary r a d i c a l s , l e t M be class of R-rings which are S-semi-simple, and l e t W be U(M). the If W i s hereditary, then W = (S:R), Proof; Since R i s hereditary and semi-simple classes are hereditary, the c l a s s M i s hereditary and so determines an upper r a d i c a l . suppose A i s a ring i n R A W. d e f i n i t i o n of W simple. g R i s closed under epimorphs, so by A can.have-no nonrrzero. »epimorph-which. is^S-semi: But then A e S, so K A such that R \ V $ S . A rj v(A) «= v ( A ) . N S , Let A be i n M. Now suppose V i s a r a d i c a l Then 0 » s(A) So A i s l/-semi-simple. Corollary; . Thus W = (S:R). Let R be a hereditary r a d i c a l and l e t W be the upper r a d i c a l generated by R, Proof; r(A) n v(A) = Since W i s the largest r a d i c a l f o r which M-rings are semi-simple, N l 7.9 First, R* « (0:R) If W i s hereditary, then W = R*. i s the upper r a d i c a l determined by R-rings which are semi-siiaple with respect to the zero r a d i c a l ; that i s , the class R, §3, The L a t t i c e of Radicals f o r Boolean Algebras Snider's r e s u l t s can be appled immediately to r a d i c a l s of Boolean r i n g s , and since Boolean r i n g r a d i c a l s are hereditary, we can use the descriptions of Prop. 7.8 and Cor. 7.9. 7.10 Proposition; The class of Boolean r i n g r a d i c a l s forms a com- plete, Brouwerian, pseudo-complemented d i s t r i b u t i v e l a t t i c e with extreme elements. If R and S are Boolean r i n g r a d i c a l s , then (S:R) i s the upper r a d i c a l generated by R-rings which are S-semi-simple, and R* i s the upper r a d i c a l generated by R. We r e c a l l that any non-zero r a d i c a l class must contain the two-element Boolean algebra. Thus R A R* •» 0 e n t a i l s that either R or R* must be 0, so that pseudo-complementation is trivial, ^However, t h i s same f a c t means that we can discard 0 and the Boolean r i n g r a d i c a l L(2) w i l l serve as a zero f o r the new l a t t i c e . In order to see that the l a t t i c e - t h e o r e t i c properties are e s s e n t i a l l y unchanged, a l l we need do i s v e r i f y the following: 7.11 Lemma: If R and S are non-zero Boolean r i n g r a d i c a l s , then (S:R) i s non-zero. Proof: Since R and S are non-zero, 2 i s i n both of them, and then R A L(2) - 1(2) $ S, Then L(2) $ (5:R). Then Prop, 7,10 holds without change f o r non-zero Boolean ring r a d i c a l s , except for the description of pseudo-comple- mentSp which now becomes R* « (L(2_):R); that i s , R* i s now the upper r a d i c a l generated by atomless R-rings, 7.12 The Isomorphism Theorem: Let Lat(E) be the class of non-zero r a d i c a l classes of Boolean rings and Lat(A) the c l a s s of Boolean algebra r a d i c a l s . Let f be the map which sends any non-zero Boolean r i n g r a d i c a l R into the class of Boolean algebras i n R. Then f i s a one-to-one correspondence between Lat(8) and Lat(A) which preserves order i n both d i r e c t i o n s , and so <Lat(B);v,A, Proof: *, L(2), 8> * <Lat(A); v , A , *, 0, A ) . By d e f i n i t i o n of r a d i c a l classes .„o.f .algebras,, f . i s onto.. Since we can recover R from f ( R ) as the c l a s s of Boolean rings, a l l of whose p r i n c i p a l ideals are i n f ( R ) , f i s one-to-one. The r e s t i s obvious. For the sake of completeness, we give a description of R* and R** f o r Boolean algebra r a d i c a l s . 7.13 Proposition: Let R be a r a d i c a l c l a s s of Boolean algebras. R* i s the class of a l l algebras with no atomless epimorphs i n R, R** Then and i s the c l a s s of algebras A such that any atomless epimorph of A has an atomless epimorph i n R. 7.14 Proof; Corollary ; If A i s atomless, then 1(A)* < U(A). If B has no atomless epimorph i n 1(A), then i t c e r t a i n l y has no p r i n c i p a l i d e a l of A as an epimorph. We now have a l l the notions required to state the conjecture mentioned i n the discussion following Prop. 6.1. The conjecture i s that i f R i s a r a d i c a l c l a s s and A » H(A^j i e I) i s a product of Ralgebras such that P(I) e R, then A.e R *. ft Recalling that P(I) i s a r e t r a c t of A, l e t M be the ideal of A such that A/M = P(I). If J i s any i d e a l of A such that A/J i s atomless, l e t K(J) be the i d e a l of A generated by M and J , The conjecture would be proved i f we could show that A/K(J) i s i n f i n i t e f o r any such J . of the Preliminaries (We use Pierce's r e s u l t 3, j ) and the d e f i n i t i o n of R**.) We also present some considerations related to the conjecture that J «= 0. epimorphs Suppose i s a c l a s s of homogeneous algebras closed under (hence containing 2) and that ^ geneous algebras. consists of a l l other homo- Then any r a d i c a l R f o r which H^-algebras are r a d i - c a l and ^ - a l g e b r a s are semi-simple must s a t i s f y L(H^) < R ^ UCf^). Taking H-^ = ( 2 ) , we see that the conjecture J = 0 i s a s p e c i a l case of the conjecture that L(H^) ° U ^ ) . One can e a s i l y extend Prop. 7.8 to show that i f S » U(M), then (S:R) = U(R n M), from which i t follows that i f R and S have the same homogeneous algebras i n them, then (J:R) » (J:S). 7.15 Thus we get: If J «• 0 and R and 5 are r a d i c a l classes with the Proposition: same homogeneous algebras i n them, then R* «= S*, §4<> Dual Atoms and Complements i n Lat(A) Snider's proof [25] that the l a t t i c e of hereditary r a d i c a l s for a s s o c i a t i v e rings has no dual atoms can be considerably s i m p l i f i e d for Boolean algebra r a d i c a l s . 7.16 Proposition: If R i s a proper r a d i c a l c l a s s (that i s , not every algebra i s r a d i c a l ) , then R i s properly contained i n a proper r a d i c a l class. Proof: Since R cannot contain a l l f r e e algebras, l e t a tea. some o r d i n a l such that F a £ R. l y containing R, Then F fl E F V R, so t h i s i s a r a d i c a l c l a s s properfl For any 8 > a, F^ i s not i n R or i n F » a Being unequi- vocal, i t i s semi-simple with respect to both r a d i c a l s , and so i t i s F a V R-semi-simple. Hence F V a R i s proper. Snider [26] gives a c h a r a c t e r i z a t i o n of complemented hereditary r a d i c a l s which we can use to deduce that 0 and A are the only complemented elements of Lat(A). following stronger 7.17 Proposition: i s proper. We choose to deduce t h i s from the result: The supremum of any set of proper r a d i c a l classes Proof: Let {R^: i e 1} be a set of r a d i c a l c l a s s e s . choose a ( i ) such that has a supremum a. t For each i e I, Since I i s a set, the set of <x(i) Then F^ i s R^- semi-simple f o r a l l i e I, and so i s semi-simple v i t h respect to the supremum of these r a d i c a l s , 7.18 Corollary: 0 and A are the only complemented elements of Lat (A), S5« Locating Known Radicals i n Lat(A) We have already obtained some l a t t i c e - t h e o r e t i c r e l a t i o n ships between our r a d i c a l s , and they w i l l not be repeated here as they are summarized i n the diagram which comprises §7. 7.19 Proposition: Proof: For a l l a, P * - ?„• F a * " Let V be any one of the following classes: atomless P -algebras, a for any a, or atomless P-algebras. Then V s a t i s f i e s the conditions of Prop, 4.8 and so the upper r a d i c a l determined by Y Is V . Q But R*, f o r any r a d i c a l R, i s the upper r a d i c a l generated by atomless R-algebras. Corollary : Lat(A), 7.20 V Q i s i n the Boolean algebra of s k e l e t a l elements of Furthermore, P « P * and f o r each a, P Proposition: Q < V* Q Let B be the least o r d i n a l such that exp Kg > exp// Then 8 i s the least o r d i n a l such that P„ •$ P . e containment i s proper. In t h i s case, the a Proof; By c a r d i n a l i t y , P does not have P^ as an epimorph, but a l l a smaller power-set algebras are epimorphs of P » If the r a d i c a l s are A equal, then V = V Q Q A » P A => P * Q Q A P = 0, which i s a contra- Q diction. 7.21 Proposition; Let g be the least o r d i n a l such that exp /{^ > exp Then f o r a l l a >, B, t> * - 0. a Proof; For any a, we have V * ? 4 t ? so that V * i P * - P .' 7 4 o . A 2 2 • « • * Yt a a Corollary; C * a Proof; = °* i V *. Q Q 0 If a > 6, then by Prop. 7.20, But then V * « 1? A P * - 0. a 0 Let B be as above, and l e t y » exp H g . Q Then f o r : This follows., from the fact ..,that .£ ra J?g« u 7.23 Proposition; Proof; F p c Let ft - fr °. g a Then C a ^ G s $ C p i The f i r s t containment was proved i n Prop. 4.6. has F a as an epimorph. by i n j e c t i v i t y of T t Note that Then i f A has F as a subalgebra, i t must, p have F a as an epimorph. This, proves the second containment. 7.24 Corollary; Let y be as i n Cor, 7,22. Then f o r a l l a £ y, G * - 0. a 7.25 Proof; Proposition: Let « exp U » By c a r d i n a l i t y , P^ E Fg, a Then ? a « Fg, 7.26 Proposition: Proof: Clearly, F 7.27 7.28 Let H e a Proof: F 1 « c a + 1 C « exp H g a + 1 . } a Proposition: •• r < a F « V . a a Corollary: *- Proof: F C a 1 tf . Then F a+1 < C Q + 1 « C p - P . Q .$ E . a i s C -semi-simple and E Q a 0 Q i s the largest r a d i c a l f o r which t h i s i s true. 7.29 Corollary: 7.30 Proposition: Proof: 7.31 0 7.32 For a > 0, E * * 0. a £ E By Cor. 7.29, f Q E T h e n Proposition: Proof: . Apply Cor. 7.14. o * * o * a« E 0 + 1 F * * E . Proposition: Proof: F F <-E E a* E a» a $ E . ± n o d s o E Then E * 3 F * . a Q By Prop. 7.30, o * " °* F * « E . Q Q By Prop. 7.30, F * •$ E Q QQ Suppose A does not have F epimorph, and l e t B be an atomless epimorph of A. If B e F as an Q f then i t has a countable p r i n c i p a l i d e a l which would be isomorphic to F , Q c o n t r a d i c t i n g the assumption on A, i n F ; that i s , E c Q <$ ¥. Q hence A has no atomless epimorph 7.33 Corollary: Proof: — 7.34 P* « By Cor. 7.27, ' i s i n the Boolean algebra of s k e l e t a l Q which i s d i s - Q 0. The meet of s k e l e t a l elements i s s k e l e t a l . but not i n E , and P c Example 3,13 7.35 Q So also i s the r a d i c a l E A P , t i n c t from E . P , and o* o Q •$ P . o o E elements of Lat (A). in V Q F Corollary: Proof: E . Q Q i s in E Q Note that F but not i n P , Q is The algebra Q of 0 i s an atomless algebra i n E A P . o o P r o p o s i t i o n : K contains ? Q and F . 0 K* * 0, If H y = exp H » D then K £ F , and f o r a l l a J. Y» F„* *• 0, Y Proof: Since P Q arid F Q are separable, the f i r s t a s s e r t i o n follows immediately. The second statement Theorem 3.6. Since every separable algebra has c a r d i n a l i t y at most H t we get that K < F . 7»36 Proof: 7.37 Corollary: Then i f a > y, K ^ F $ « F a n d If y i s as i n Prop. 7#35* and a £ y Use Prop, 7,26 Corollary: i s simply a re-statenent of If and Prop. 7,35. F + rv* K* = A» 1 » then T h i s sharpens Cor. 7.22. •= exp ft , then C * = 0 £<$n? a l l a £ 2. that i n t h i s case, C. » P , so Cor. 7.36 a Note i s a best p o s s i b l e r e s u l t . §6. Atoms i n the L a t t i c e 7.38 Theorem: Let R be an atom i n Lat(A), Then: i ) f o r any non-superatomic algebra A e R , R = 1(A), i i ) there i s an atomless. separable algebra A i n R , such that R =• L(A), i i i ) any atomless algebra A i n R i s unequivocal, and i v ) f o r any atomless algebra A i n R , L(A)* « U(A) so that U(A)* $ 0. Conversely, i f A i s an atomless unequivocal algebra such that U(A)* & 0, then L (A) i s an atom i n the l a t t i c e of r a d i c a l s . Proof: The f i r s t assertion i s obvious, and i i ) follows from the f a c t that any..~no.n-8uper,a.tgmlc.,..algebra.*has„an.s,t,Pml.es.s .«separable«epimo,rph. 6! Using Prop. 2,15, i i i ) follows from i ) . If A i s an atomless R-algebra, then 1(A)* « U(A) by Cor. 7.14. Since L(A) i s an atom, e i t h e r 1(A) « ti(A) or L(A) A U(A) - 0. The f i r s t i s impossible, and the second implies U (A) < 1(A)*. Then i v ) follows. suppose A i s unequivocal and U(A)* ^ 0. For the converse, If A were U(A)*-semi-simple, we would have U (A)* ^ U(A), contradicting U(A)* £ 0. Thus, since i t i s unequivocal, A E U(A)* and L(A) « U(A)*, so U(A) £ U ( A ) * * ^ 1 ( A ) * ^ U ( A ) . Hence U (A) « L(A)*. Now l e t -S be any r a d i c a l and suppose A t S. A i s S-semi-simple, BO S 4 U (A) » 1(A)* and L (A) A S =» 0, Then 7.39 Theorem: P and F_ are atoms i n Lat (A). o o 1 Proof: Write P = L ( F ) o o' vocal. Also U ( F ) * * V * 1 N q Q l a s t theorem, ? Q and F and F o N Q » L(F ). o Then F o i 0, and U ( F ) * «= E * Q D and F o 0. are unequiM Hence, by the are atoms. §7« A Diagram of the L a t t i c e On the next page, we present a diagram of L a t ( A ) , which summarizes the r e s u l t s of the l a s t tVo sections. For s i m p l i c i t y , we assume GCH and we omit mention of the G^, which, by Prop. 7.23, are interspersed among the chain {C }. BIBLIOGRAPHY 1. Anderson, T., Divinsky, A., and Sulinsky, A., "Hereditary r a d i c a l s i n associative and a l t e r n a t i v e rings," Canad. J . Math. 17 (1965), 594-603. 2. Amitsur, S. A., "A general theory of r a d i c a l s . I I : Radicals i n rings and bicategories," Amer. J . Mathsi 76 (1954), 100125. 3. Armendariz, E. P., "Closure properties i n r a d i c a l theory," P a c i f i c J . Math. 2£ 5(1968), 1-7. ; 4. Birkhoff, G,, L a t t i c e Theory, 3rd e d i t i o n , American Mathema t i c a l Society, Providence, R. I., 1967. 5. Cramer, T., "Extensions of free Boolean algebras," Proc. Lond. Math. Soc. (to appear)• 6. Day, G. W., "Free complete extensions of Boolean algebras," P a c i f i c J . Math. 15 (1965), 1145-1151. 7. — •,*" Super atomic Boolean algebras," P a c i f i c J . Math. 23 (1967), 479-489. 8. Divinsky, N,, Rings and Radicals, U n i v e r s i t y of Toronto Press, Toronto, 1965. 9. Easton, W, B., "Powers of regular c a r d i n a l s , " Annals of Math. Logic 1 (1970), 139-178. 10. Efimov, B., "Extremally disconnected bicompacta," Dokl, Akad. Nauk SSSR 172 (1967) • Soviet Math. Dokl. 8 (1967), 168-171. 11. , "On embedding of Stone-Cech compactifications of d i s crete spaces i n bicompacta," Dokl. Akad. Nauk SSSR 189 (1969) = Soviet Math. Dokl. 10 (1969), 1391-1394. 12. Gillman, L., and J e r i s o n , M„, Rings of Continuous Functions, Van Nostrand, Princeton, N. J , , 1960, 13. Gratzer, G., "Homogeneous Boolean algebras," Notices Amer. Math. Soc. 20 (1973), A-565. 14. Halmos, P., Lectures on Boolean Algebras, Van Nostrand Mathemat i c a l Studies #1, Princeton, N. J . , 1960, 15. Kausdorff, F., "Uber zwel Satze von G. Fichtenholz und L. Kantorovitch," Studia Math. 6 (1936), 18-19. ifeo Horn, A,, and T a r s k i , A., "Measures i n Boolean algebras," Trans. Amer. Math. Soc. 64 (1948), 467-497. 17, Kurosh, A, G,, "Radicals of rings and algebras," Mat. Sbornik 33 (1953), 13-26. 18„ Lakser, H«, "The structure of pseudo-complemented d i s t r i b u t i v e l a t t i c e s , I: Subdirect decomposition," Trans, AmerAcSoc. 156 (1971) , 335-342. 19. L e a v i t t , W. G., "Sets of r a d i c a l classes," Publ, Math. Debrecen. 14 (1967), 321-324. 20. Monk, J , D., and Solovay, R,, M,, "On the number of complete Boolean algebras," Algebra U n i v e r s a l i s 2 (1972), 365-368. 21. Mostovski, A., and T a r s k i , A., "Boolesche Ringe mit geordneter Basis," Fund. Math. 32 (1939), 69-86. 22. Pierce, R. S., "A note on complete Boolean algebras," Proc. Amer. Math. Soc. 9 (1958), 892-896. -23. « — — • V Some ^questions *about"comlple te Boolean 'algebra s," Proc. Symp. Pure Math. 2 (1961), 129-140. f 24. S i k o r s k i , Roman, Boolean Algebras, 3rd edition, Springer-Verlag New York Inc., 1969. 25. Snider, R. L,, " L a t t i c e s of r a d i c a l s , " P a c i f i c J . Math. 40 (1972) , 207-220. 26. —-— , "Complemented hereditary r a d i c a l s , " B u l l . A u s t r a l . Math. Soc. 4 (1971), 307-320. 27. Yaqub, F. M., "Free extensions of Boolean algebras," P a c i f i c J , Math. 13 (1963), 761-771. \ INDEX OF NOTATION Page references are given where they might be h e l p f u l . Ordinals: a, 8, Yt"... Well-ordered chains: { l } (29), (h (A)} (29), a a {R > (63). a Cardinals: K, , |A|; K+ = next l a r g e s t c a r d i n a l a f t e r K; exp K •= 2*.. Rings and Algebras: A, B, C, ... ; A = p r i n c i p a l i d e a l of A generated by x; h*(A) (29); A « normal completion of the algebra A (10-11); 2_ =• two-element Boolean algebra; F on K q = f r e e algebra generators; P_ «= power-set algebra on a set of cardin- - a l i t y - / ^ ; "Q (49). Topological Spaces: X, Y., Z, ... ; gX • Stone-Cech compactification of the space X; 8N m Stone-Cech compactification of a countable set with the d i s c r e t e topology; 2 "'** product of K copies of K H the two-element discrete space, a Cantor space; 2 ° the Cantor s e t ; S(A), S(x), S(I) « concepts associated with the Stone duality (6). Constructions: Product of algebras: H(A^: i e I ) , A x B, A K (8); weak product of algebras: wIl(A^: i e I) (8); coproduct of algeb- r a s : E(A : i e I ) , A + B (12), KA (75); product of t o p l o g i - c a l spaces: II(X : i e I ) , X . < ± L a t t i c e operations: (b:a), a* (78-79). Cardinal properties: v (51); p (63); n, 6(64); c(R) (76). Classes of Rings and Algebras: X, M, H, ... ; r a d i c a l classes with corresponding r a d i c a l i d e a l : R, r ( A ) , S, s(A), ... (18-19); lower r a d i c a l : L(X) (28); upper r a d i c a l U (/) (45); 8 - the c l a s s of Boolean rings; A • the class of Boolean algebras; Lat(B) « the class of r a d i c a l classes of Boolean rings; Lat(A) « the class of r a d i c a l classes of Boolean algebras. Radical Classes of Boolean Algebras: 0 ** the superatomic Boolean algebras (36), F a = lower r a d i c a l generated by F (38), & K » lower r a d i c a l generated by separable algebras (40), P Q • lower r a d i c a l generated by P (40), a P = lower r a d i c a l generated by complete algebras (42), E Q => upper r a d i c a l determined by F (48), G a = upper r a d i c a l determined by F (48), a Q J <= upper r a d i c a l determined by atomless homogeneous algebras (50), J v «= upper r a d i c a l determined by atomless v-homogenenous algebras (51), C Q V A » algebras without F " algebras without P a as a subalgebra (53), as an epimorph (54).
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Radical classes of Boolean algebras Galay, Theodore Alexander 1974
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Title | Radical classes of Boolean algebras |
Creator |
Galay, Theodore Alexander |
Date Issued | 1974 |
Description | This thesis obtains information about Boolean algebras by means of the radical concept. One group of results revolves about the concept, theorems, and constructions of general radical theory. We obtain some subdirect product representations by methods suggested by the theory. A large number of specific radicals are defined, and their properties and inter-relationships are examined. This provides a natural frame-work for results describing what epimorphs an algebra can have. Some new results of this nature are obtained in the process. Finally, a contribution is made to the structure theory of complete Boolean algebras. Product decomposition theorems are obtained, some of which make use of chains of radical classes. |
Subject |
Algebra, Boolean |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-01 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080527 |
URI | http://hdl.handle.net/2429/19509 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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