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Radical classes of Boolean algebras Galay, Theodore Alexander 1974

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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 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. an advanced degree at the University of British Columbia, I agree that Department of The University of British Columbia Vancouver 8, Canada Date CCt^^t^LZ^ /S~ / f7? ASSTRACT 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 suggest-ed 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. TABLE OF CONTENTS INTRO DUCT ION oe»*fle»Qecoee>eee«O0eo9oeoo*OQ&po»»«o6»«0e»*oe0«e»ee I PRELIMINARIES ........ coco o » BO 9»e«ooeoce » » o • • e 0 o o e e o * o c © e« e • • • • CHAPTER ONE: RADICAL CLASSES OF BOOLEAN ALGEBRAS §1 » GenSX'Sl Rad XCal Ill&Oir V oe.cti.»«eooo«oo.e.0»e«.vo.o«o.e \& § 2 . Radical Classes of Boolean Rings „..... ...... . , . . . . . 21 §3. Radical Classes of Boolean Algebras .••.«»«•••••.>.«.o 23 §4» Conventions and Summary OIMIKMI 26 CHAPTER TWO: THE LOWER RADICAL 51. The Lower Radical Construction .....o..o*.o........... 28 § 2 . The Lower Radical for Boolean Algebra Radicals . 30 §3 c T ll£ SilpGTTfitlOIIiiC R&d1C&1 0o?««eoo«eo«o«eo*eoe6 0 a « 6 o o e « « 33 §4 e The Cardinality Radicals •••» o e e«oo«* o e»D 0«e» e«eo«e»«* 38 5 5 o T llS POWGJT^Sst Radical S oof>OQ«o«>6ceoo»?**o*«ee«O0*eoe*'O 40 CHAPTER THREE: THE UPPER RADICAL § 1 . The Upper Radical Construction ........<>.............. 44 52. The Characterization Theorem . . . . . r 45 § 3 . Atomless Boolean Algebras ............................ 46 §4. Some Upper Radicals .................................. 47 § 5 . The Upper Radical Determined by Homogeneous Algebras 50 CHAPTER FOUR: CRAMER'S RADICALS § 1 . . The Classes C a 00000 .0000 . .09, 0 . 0 . . . . 0 . 0 0 0 0 000 0000 « • o o 53 92. -The as Upper Radicals •»0v6e««*oo«o«eooe««««««e9*e« 54 §3 , The Radical Vn c o o . . . . . . . . e o 00 ee te 00 o © o o ti o © •» 0 0 00 00 00 55 CHAPTER FIVE: DECOMPOSITIONS OF COMPLETE ALGEBRAS § 1 0 Xl)2 GCHSITGLI SGttinf^ 00000000000000000 00000000000000 00 o 37 52, Decompositions into Homogeneous Algebras . 59 § 3 . Decompositions into Unequivocal Algebras •....•.»«••.. 61 §4, Connections with Cardinal Properties «•.....•....•..•. 63 CHAPTER SIX: CLOSURE PROPERTIES OF RADICAL AND SEMI-SIMPLE CLASSES SI. Closure of Radical Classes under Products v..».«•«.... 67 J.2. .Closure,of.Radical j.Clas.see,,undsr ,C,QpXQdu,cts ...„.,............. 69 § 3 . Coproducts and Semi-Siroplicity 0 0 0 . 0 0 . 0 0 . 0 0 0 . . . 0 0 . 0 0 0 . 75 00000 83 o o •«0 83 CHAPTER SEVEN: THE LATTICE OF RADICALS §1. Lattice-Theoretic Preliminaries 0, . 0 0 . . . . 0 0 0 0 . 0 . 0 79 §2. The Lattice of Radicals for Associative Rings §3. The Lattice of Radicals for Boolean Algebras . §4 . Dual Atoms and Complements in Lat(A) §5 . Locating Known Radicals in Lat(A) §6 . Atoms in the Lattice .........o...0000 .000000 . §7. A Diagram of the Lattice ..... BIBLIOGRAPHY > Q C 0 « • « O > « M OOOO 0 e 0*000* 0 • 00*00000 ee 00 0 * • « 0000*0 0*0000 « 0 e o o < INDEX OF NOTATION . . . . . . . . . . . . . . . . . . . . O 0 0 0O0OO0OO0O00 000 • 0000 88 89 93 94 00000 1 * ~ 00000 96 • 0 0 0 0 W 0 0 0 0 0 0 0 98 ACKNOWLEDGEMENTS I wish to express my appreciation to my thesis super-visor,, Dr. To Crauier;, for his continual encouragement and help during the preparation of this thesis. My sincere thanks also go to Dr c A, Adler for his friendly support and constant willingness to enter into discussion, I would al-so l i k e to thank the members of the Radical Theory Seminar, Dr. N, Divinsky, Dr c 1, Anderson9 and Dr, M, Slater, INTRODUCTION The f i r s t significant breakthrough i n the study of struc-ture by means of radicals was Cartan's classification of the f i n i t e -dimensional semi-simple Lie algebras over the f i e l d of complex num-bers. Early i n the twentieth century, Wedderburn obtained his struc-ture theorem for finite-dimensional associative algebras. It took nearly forty years for the next major development: the definition of the Jacobson radical and the density theorem. The general procedure in these cases was the same: for a class of rings„ define an ideal for each ring i n the class, c a l l i t the radi c a l p and see what can be said about radical-free (semi-simple) rings. In the early 1950's, Kurosh and Amitsur defined the general concept of a radical class, which became the subject of much subsequent research. In 1939, Mostowski and Tarski introduced the notion of a superatoiaic Boolean algebra* A number of significant results con-cerning this class of algebras have been developed since, without exploiting the fact that i t i s a radical class. In 1972, Cramer gener-alized the superatomic radical to obtain a transfinite chain of radi-cal classes of Boolean algebras. Again, the ideas and methods were not radical-theoretic i n nature. The aim of this thesis i s to obtain information about Boolean algebras by means of the radical concept, and to place these results in the setting of general radical theory. In general radical theory, i t Is necessary to begin with a class of rings closed under ideals and epimorphs. The class of Boolean algebras i s not such a class: an ideal of a Boolean algebra i s not a Boolean algebra. In Chapter Gne„ starting with the more general concept of a Boolean ring p we provide a natural definition of a radical class of Boolean algebras. In the process, we obtain two interesting results conceniing these radicals: f i r s t , every non-zero radical class of Boolean algebras contains the class of superatouiic algebras; secondly, every radical class of Boolean algebras i s hereditary. In Chapter Two, we investigate the construction of the lower radical. In general,, this i s an i n f i n i t e procedure; for Boolean algebras, the construction terminates at the second stage. If the generating class i s closed under epimorphs, we find that a complete radical algebra srust be a product of generating algebras. We define some lovrer radicals, the most important of which we c a l l power-set radicals. For any power-set radical, we obtain necessary and s u f f i c -ient conditions for a complete algebra to be in the radical, and for an algebra to generate the radical. The lat t e r result proves useful i n obtaining a product decomposition theorem for certain complete algebras. In Chapter Three, we study the upper radical construction. The algebras which are semi-simple with respect to an upper radical are characterized as subdirect products of the algebras which deter-mine the radical. An upper radical description of the superatoiaic radical yields a characterization of atomless Boolean algebras as subdirect products of separable, atomless algebras. Other upper radicals are defined, and we obtain a subdirect representation for the complete, atomless algebras. Chapter Four concerns the radical defined by Cramer. Radical-theoretic methods are used to prove and extend some of his results. In particular, an upper radical description i s obtained for some of his radicals. Pierce has conjectured that any complete Boolean algebra i s a product of homogeneous algebras. Chapter Five gives some partial results i n this direction. It i s shown that i f any descending chain of principal ideals in a complete algebra has only f i n i t e l y many isomorphism types, then the algebra i s a product of homogeneous algeb-ras. The main result asserts that certain complete algebras are products of unequivocal algebras (that i s , algebras which must be either radical or semi-simple with respect to any radical class). Some of these results are re-stated in the language of cardinal prop-erties, which provides some additional insight. Chapter Six gives some closure properties of radical and semi-simple classes under the formation of products and coproducts. Not unexpectedly, power-set radicals are shown to be closed under suitably restricted products of complete algebras. A new product i s defined which yields a radical algebra whenever a l l the algebras involved i n the construction are radical, A number of radical classes are shown to be closed under f i n i t e coproducts. We obtain two results indicat-ing that coproducts are strongly related to semi-simplicity. F i r s t , any coproduct of algebras, one of which i s semi-simple, i s i t s e l f semi-simple. Secondly, for any radical class, there i s a cardinal K such that for any algebra A of more than two elements, the coproduct of at least K copies of A w i l l be semi-simple. In Chapter Seven, we regard radical classes as elements of a l a t t i c e . The structure of this l a t t i c e is investigated. The theme .of Chapter*One i s re-iterated by.showln.the-isomorphism-of the l a t t i c e of Boolean ring radicals with the l a t t i c e of Boolean algebra radicals. Finally, we focus on the specific radicals defined in this work, with a view to locating them in the l a t t i c e . PRELIMINARIES This section outlines the terminology, notation, and basic facts to be used. Basic references are Sikorski [24] and Halmos [13]. Further references are given as needed, and details are provided for results not easily accessible or e x p l i c i t l y stated i n the literature. §1 0 Fundamental Notions a) We assume familiarity with the concept of Boolean ring, both i n the ring-theoretic and lattice-theoretic settings. Vie w i l l use both the ring operations (+, • ) , and the l a t t i c e operations of join ( v ) , meet% ( a ) , and complementation ('). The concepts of ideal in the two settings coincide. We w i l l use the notation A x for the principal  ideal of the ring A generated by the element x. The term Boolean algebra (or...sinrply algebra) w i l l be used for a Boolean ring with a unity 1 distinct from the zero 0. A subalgebra of a Boolean algebra i s a subring containing the unity. Any ideal of a Boolean algebra gen-erates a subalgebra, consisting of the ideal together with the comple-ments of elements of the ideal. Such a subalgebra w i l l be called an ideal-generated subalgebra. 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 principal ideal of a Boolean ring i s a Boolean algebra, i i ) any Boolean ring can be embedded as a maximal ideal i n a Boolean algebra, which is unique up to isomorphism, and i i i ) any non-zero epimorph of a Boolean algebra is 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 for the class of a l l Boolean algebras, and the symbol B for the class of a l l Boolean rings, d) We assume familiarity, also, with the Stone duality theory, which assigns to any Boolean algebra A, a topological space S(A), called i t s Stone space, which is Bausdorff, compact, and totally disconnected. The correspondence is reversible, and allows the following interchange of algebraic and topological concepts: i) an element x of A (or the principal ideal A x) corresponds to a clopen subset S(x) of S(A), and A is 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); namely6 S(I) i s the union of the clopen sets S(x) for x e I; i i i ) the epimorph A/1 of A corresponds to the closed subset S(A) - S(I) of S(A); iv) an embedding A >-> B corresponds to a continuous sur-jection S(B) —» S(A); v) an epimorphism A — » E corresponds to a continuous injec-tion S(B)^-> S(A), e) Lattices w i l l always be assumed to have extreme elements 0 and 1„ which are di s t i n c t . An atom in 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 distinct from 1 which i s contained only in 1 and i t s e l f . In the Stone space of a Boolean algebra, atoms appear as isolated points. Elements x and y of a l a t t i c e are disjoint If x A y • 0, A set D is disjointed i f any two distinct elements of i t are disjoint. The supreinutn of an arbitrary E i n a l a t t i c e , when i t exists, w i l l be denoted by sup E. The terms "•complete" (a-cotriplete)'- w i l l -b'fer u'se'd "to -indicate 1 thafarb iferary - ('count-able) suprema always exist. 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+ for the next largest cardinal, and exp K for the cardinal more commonly denoted by 2 K . 52* Properties o£_ Boolean Rings and Algebras a) If I Is an ideal of a Boolean ring As and J i s an ideal of I, then J i s an ideal of A. b) Every non-zero Boolean ring has a two-element epimorph. We w i l l denote the two-element Boolean algebra by 2. c) For any collection {A^ i e 1} of Boolean rings (algebras), the Cartesian product of the underlying sets together with the point-wise operations forms a Boolean ring (algebra), which we c a l l the product ed by x A2 s « " x A n„ The Stone space of the product of algebras i s the Stone-Cech compactificatioa of the disjoint union of the S(A^), d) The subset of the product A of algebras A^, consisting of elements which are 0 in a l l but a f i n i t e number of coordinates i s an ideal 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), Its Stone space i s the one-point compactifi-cation of the disjoint union of the S(A^). e) An algebra A i s a retract 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 composi-tion gf i s the identity on B. Any principal ideal of an algebra A i s a retract of A, In particular, any factor algebra i n a product or weak product A i s a principal ideal of A and so i s a retract of A, of the A^ and denote by A 0 2 n(A^s i e I). Finite products are denot-f) Any Boolean ring A admits a product decomposition at any element x: namely, A is the product of A with the ideal of elements disjoint from x. If A is an algebra, we have A - k x A„«, x g) A subset D of a Boolean algebra A is said to be dense i f i t con-sists of non-zero elements, and any non-zero element of A contains an element of D, In particular, 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 cardinality at most exp H Q. Furthermore, there i s only one complete, atomless, separable algebra: the quotient of the algebra of Borel sets of real 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 AH I = 0. The restriction of the natural epimorphism f: B — » B/I to A is 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 is s t r i c t l y larger than I. By the maximality of I, there is a non-zero element a in A fl J. Being in J, this element must have the form a = x v c, where x e I and c S b, Since f is an embedding, 0 ^ f(a) m f (c) ^ £ (b). Hence the copy of A in B/I is denser S3 0 Comgljct^ Algebras a) The algebra of a l l subsets of a set X w i l l be denoted by P(X)„ and is called a power-set algebra. It i s characterized as the product of | x | copies of the two-element algebra, and i t s Stone space is the Stone-Cech compactiflcation gX of the set X with the discrete topology. When |x | = H a i s i n f i n i t e , we w i l l denote P (X) by P a. b) Any product of algebras over an index set I has P(I) as a retract, c) Complete algebras are precisely the retracts of power-set algebras. They'can also be characterized as 'the •Inject lye "Boolean algebras. An algebra C i s said to be injective i f , g A > > B whenever f is a horaomorphism from A to / £ / C and g i s an embedding of A in B, then C 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 cardinal Ha i s the cardinal-i t y of a complete Boolean algebra i f and only i f Ha ° c'U0i» e) Any 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. principal ideal of A generated by x. Hence x x ° f) An algebra A satisfies the countable chain condition (c,c.c) i f any disjointed subset of A i s at most countable. Proposition 2_s If A i s an i n f i n i t e algebra satisfying c . c . c , then |A| - |A| H°. Proof: Every element of A can be represented as a disjoint (hence, at most countable) supremuro of elements from A. Hence | A | < | A | ° , Also, |A| ° 4 |A| ° » |A|, by d) above. g) Any o-complete algebra s a t i s f i e s the following condition: If the algebra A is isomorphic to i t s principal ideal A , then i t i s also isomorphic to A^ for 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 principal 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: Let A be homogeneous and suppose x i s a non-zero element of A, Choose some non-zero y in 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 - A. j ) Pierce (see Cramer [5]) has shown that any in f i n i t e epimorph of a o-algebra has P Q as a retract. a) If i s the Stone space of A^ for 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^ in A. We identify 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 collection of non-zero elements x^ chosen from sub-algebras with different 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 , in fact, a retract of A 0 An easy topological argu-ment yields a stronger result; i f J is a subset of I and is a retract of A. for each j e J , then £(B.; j e J ) i s a retract of A. In particular, £(AJ: j e J ) i s a retract of A„ We assume these partial coproducts are actually contained i n A» Under this 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^ for each i e I, then H(F^t i e I) i s closed in fl(X^: i e I), We spell this out algebraically: Proposition 4_: Let A « Z(A^i i e I), and l e t be an ideal of A^ for each i e I. Let J be the ideal of A generated by the union of the J±* Then: i) "J consists of a l l f i n i t e joins 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 ideal i i ) A/J - L ^ / J ^ : i e I). The proof i s a straight-forward ve r i f i c a t i o n . The proposi-tion has some useful corollaries: i ) Let I be an ideal of A, J an ideal of B, and l e t K be the ideal of A + B consisting of f i n i t e joins 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 joins 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 ) f l A b - k& + B b, e) 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^ for each i £ I. A subalgebra B of A i s called a subdirect  product of the A^ i f each of these epimorphiems, restricted to B, s t i l l maps onto A^. b) If { i e 1} i s a collection of ideals of an algebra A whose intersection i s the zero ideal, then A can be represented as a sub-direct "produc t of the 'K/'J^, §6. Free Algebras a) F i s a free algebra on K generators i f F i s generated by a set X of cardinality 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) The free algebra on K generators can be realized as the coproduct of K copies of the four-element Boolean algebra, or equivalently, as K the algebra of clopen subsets of the Cantor space 2 , the topological product of K copies of the two-element discrete space. c) For i n f i n i t e free algebras, the cardinality of F Q i s B a . Further-more, F p i s homogeneous and satisfies c,c,c. Hence i t s normal comple-tion F a i s homogeneous and has cardinality H ^ 0 , d) 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 retract of A, e) The countable free algebra F q i s the only countable, atomless Boolean algebra. Its normal completion F i s complete, atomless, and separable, and so i t is 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 for any homomorphism f of P to 'an algebra *B, '"'ait'd -any "epimorphism g of A onto B, there i s a homomor-P A V? B phism h of P to A such that gh = f, §7, Universal Mapping Properties a) Every algebra i s an epimorph of a free algebra. There i s an analogous result for complete algebras: Proposition 5_: If A i s a complete algebra of cardinality at most fo^, then A i s an epimorph of F^, Proof: A i s an epimorph of F a , which can be embedded in F a, Then in j e c t i v i t y of A yields the result. A form of this result was f i r s t proved in Efimov [10] by a f a i r l y involved topological argument. b) Kausdorff [15] has shown that i f = exp/"/a, then Fg can be embedded in P^. This has some important consequences. Proposition 6: Let H » exp ft . Then F„ i s a subalgebra of a Boolean — _ j — _ g a . 8 algebra A i f and only i f P^ i s an epimorph of A. Proof: One direction follows from the i n j e c t i v i t y of P a and the fact that i t i s an epimorph of Fg. The other direction 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; Let exp Ha° Then Fg i s a retract of P^. Proof: Since P a is complete and Fg i s a subalgebra of i t , we get that F i s also a subalgebra of i t . But any complete subalgebra of p an algebra i s , in fact, a retract of that algebra. c) The most useful form of the preceding results for our purposes i s the following: Proposition jj: Let Kg » exp f/ a. Then P a and Fg are epimorphs of one another. Corollary; Any two complete algebras of cardinality exp ft are epi-morphs of one another. Proof; It suffices to show that i f A i s complete of cardinality// = exp HQt then A and P Q are epimorphs of one another. By Prop. 5, A is an epimorph of F arid hence of P Q, by Prop, 8, By Pierce's result (§38 j ) , P Q i s an epimorph of A. d) If a < £, then P a i s a principal ideal (hence a retract) of Pg, The algebras are not isomorphic since they do not have the same number of atoms. They might, however, have the same cardinality. It i s consistent with the usual axioms of set theory to assume, for example, that exp ft = exp ft(see Easton [9]). It i s also consistent >to-»assutna--..t'hafc=,tt-< ;-(>.*lmpMes«ex<p-/tya'^  quence of GCH, In any case, we have the following: Proposition 9; t P a and P^ are epimorphs of one another i f and only i f exp Ha = exp ft^0 Proof: Assume exp fta a exp ft^ •= H^* By Prop. 8» P a and F^ are epi-morphs of one another, as are Pg and F^, Hence, the two power-set algebras are epimorphs of one another. The other direction i s clear, e) Pierce's result (§3, j) has useful 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 free epi-morphs. In particular, 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 radical class of rings i s veil-known and has been studied extensively. The theory can be applied immediately to the class of Boolean rings, but some adjustment in the definitions and results i s necessary for the class of Boolean algebras, §1, General R.adical Theory This section i s a review of the basic concepts of radical •'theory, f orv-sssociaftive* rings, • - A 'general '.reference «f or this material i s Divinsky [8], 1.1 Definition: A class of associative rings i s called universal i f i t i s closed under the formation of epimorphs and ideals. In what follows, we assume that a l l classes of rings con-sidered are subclasses of some fixed universal class, 1.2 Definition (Amitsur [2], Kurosh [17]): A non-empty class R of rings i s a radical class (or simply a radical) i f i t satisfies the following properties; i ) every epimorph of an R-ring i s an R-ring, i i ) every ring A contains an R-ideal, called the R-radical of A and denoted by r(A), which contains every R-ideal of A, and i i i ) for any ring A, r(A/r(A)) = 0. 1„3 Definition; If R i s a radical class and A i s a ring such that r(A) » 0, then A i s called R-seal-simple. When some fixed radical class i s being discussed and there i s no danger of ambiguity,, we w i l l simply use the terms "radical" and "semi-simple" without specific reference to the radical class,, It i s obvious that the t r i v i a l ring {0} i s the only ring which can be simultaneously radical and semi-simple with respect to a radical class. 1.4 Proposition; For any radical class R and any ring A, r(A) i s the intersection of a l l ideals I of A for which A/I i s R-semi-simple. The following propositions characterize radical classes and give some of their closure properties. 1.5 Proposition; A class R is a radical class i f and only i f : i ) R i s closed under epimorphs, and i i ) i f A i s a ring 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 radical class 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 Ae 1.7 Corollary; If R i s a radical class, then the weak direct product of R-rings i s an R-ring, 1.8 Proposition: If R i s a radical class and I i s an R-ideal of a ring A such that A/I i s in R, then A i s in R, 1.9 Def inition: For any class M of rings, we say a ring A i s an approximate W-ring^ i f every non-zero ideal of A has a non-zero epi-morph in M, 1.10 , Proposition; A c l a s s l o f .rings M is. .thexlass of .all R-semi-simple rings for some radical class R i f and only If M i s equal to the class of a l l approximate M-rings. In this case, R can be recov-ered from M as the class of a l l rings with no non-zero epimorph in M, 1.11 Def initi o n ; A class of rings i s called hereditary i f every . ideal.of a ring in the class i s also i n the class, 1.12 Proposition (Armanderiz [3]): A class M i s the class of a l l semi-simple rings for some radical class 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 is an M-ideal of a ring A for which A/I i s in M, then A i s in M, and iv) i f I is an ideal of a ring A such that I/B i s a non-zero M-ririg for some ideal B.of I, then there i s an ideal C of A contained i n I such that I/C i s a non-zero M-ring. The fact that any semi-simple class of associative rings is hereditary was f i r s t proved by Anderson, Divinsky, and Sulinski [1], 1.13 Proposition; A radical R i s hereditary i f and only i f r(I) I H r(A) for any ring A and any ideal I of A. Furthermore, i f R i s hereditary, then for any ring A, r(A) i s the ideal of A generated by the principal R-ideals of A0 1.14 Definition; For any class C of rings and any ring A, define c(A) to be the ideal of A generated by the principal C-ideals of A. This coincides with the definition of r(A) for a radical class R provided that R i s hereditary. In the ©ext section, we show that a l l radicals 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 universal class, so we can immediately apply the concepts and results 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 radical class of Boolean rings, then the two-element Boolean algebra 2^  i s in R. Proof: Any non-zero Boolean ring has 2_ as an epimorph. 1.16 Corollary; If R is a non-zero radical class of Boolean rings, and A i s a Boolean ring with a maximal ideal I in R, then A i s in R, Proof: Both I and A/I - 2_ are in R. By Prop, 1.8, A i s in R. 1»17 Proposition: Every radical class of Boolean rings i s hereditary. Proof: Let A be in the radical class R of Boolean rings. Any ideal I of A i s generated by the principal ideals of A contained within i t . "Each of'these'is an epimorph b'f 'A and so'is'in'R, ""By'Pro'p. '1,*'6, I i s in R. We are now ready to prove the theorem which allows us to restrict attention to Boolean algebras, 1.18 Theorem: Let R be a non-zero radical class of Boolean rings, and l e t S ° A fi R, the class of Boolean algebras in R, Then for any Boolean ring B, the following are equivalent: i ) B i s in R, i i ) every non-zero principal ideal of B i s in 5, iii)..C i s in S, where C is the Boolean algebra containing B as a maximal ideal. Froof; If B is in R, then every non-zero principal of B i s in R by Prop. 1.17,, and C i s in R by Cor. 1.16. Being algebras, they are, in fact, in S. Thus i) implies i i ) and i i i ) . If i i ) holds, then every principal ideal of B is in R. Since B i s generated by i t s principal ideas, i t i s in R by Prop. 1.6. Thus i i ) implies i ) . If i i i ) holds, then C i s in 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 Definition; A non-empty class 5 of Boolean algebras i s called a radical class i f and only i f there i s a radical class R of Boolean rings such that S = A ft R, The remainder of this section shows that a l l the concepts and results of radical theory can be expressed (with only minor modi-fications) entirely in the language of Boolean algebras. In keeping with Defn. 1,14, for any class S of algebras, the ideal s(A) of the algebra A i s the ideal generated by the S-ideals (necessarily principal) of A, 1.20 Proposition: A class S of Boolean algebras i s a radical class i f and only i f : i ) S i s closed under algebra epimorphs, and i i ) s(A/s(A)) = 0 for every Boolean algebra A. Proof: If S i s a radical class of algebras obtained from the Boolean ring radical R, then i t i s clear that s(A) «* r(A) for any algebra A, Hence, i i ) follows immediately. Also, any algebra eplmorphism i s a ring epimorphism, so i ) follows. Suppose S satisfies i) and i i ) , and l e t R be the class of a l l Boolean rings A satisfying Aj.eS for every non-zero x e A, It is clear that S i s precisely the class of a l l algebras in R, so a l l we need show i s that R i s a radical class of rings. Suppose that B i s a ring epimorph of the R-ring A, Then any non-zero principal ideal of B i s a ring epimorph of some non-zero principal ideal 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 , in fact, an algebra eplmorphism. By I ) , then, every non-zero K.principal ideal of B i s an .S-algebra, and so 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 in 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 x/s(A x) has an S-ideal, contradicting condition i i ) . By Prop, 1.5, R i s a radical class of Boolean rings. 1»21 Proposition: A class 5 of Boolean algebras i s a radical class 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 in 5. ' Proof: Suppose S i s a class satisfying i ) and i i ) , define R as in the last proposition, and repeat the argument showing that R i s closed under ring epimorphs. Now l e t A be a ring such that every non-zero ring epimorph of A has a non-zero R-ideal. In particular, every non-zero. • . ..._ ---j.- _ •• . principal ideal of A satisfies this condition, and so has an S-ideal. By i i ) , the, every non-zero principal ideal of A i s in 5, and so A i s in R. Thus, R i s a radical class of Boolean rings and S consists of a l l the algebras in i t . The opposite direction i s clear. 1.22 Proposition; Let S be a radical class of algebras, obtained from the Boolean ring radical R, Then for any algebra (ring) A; s(A) = r(A) «= {x: A, e R} » {x; x = 0 or A„ e S}„ x x ..Proof: ...Let..I « {.xs.-Ax~e*-R} , ..,A11 that,.needs,proof i s that X i s an ideal of A. Since R i s hereditary, I i s closed under subelements. The fact that I i s closed under f i n i t e joins follows from the fact that A i s generated by A and A , and Prop, 1.6, x v y x y* 1 This proposition makes precise what we w i l l mean by the expression: the radical of an algebra consists of i t s radical elements. 1.23 Definition; If S i s a radical class of algebras, we say an algebra A is S-semi-simple i f s(A) » 0. A class of algebras i s a_ semi- simple class i f i t consists of a l l S-semi-simple algebras for some radical class S of algebras. 1.24 Definition; For any class M of algebras, we say an algebra A i s an approximate M-algebra i f every non-zero principal ideal of A has an (algebra) epimorph in M. 1.25 Proposition; A class of algebras U i s a semi-simple class i f and only i f M i s equal to the class of a l l approximate M-algebras. In this case, the radical class associated with M i s the class of a l l algebras with no (algebra) epimorph in M. Proof: A straight-forward ve 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 definitions of class properties w i l l be modified i n the obvious manner. For example: 1.26 Definition: A class of algebras i s said to be hereditary i f i t Is closed under the formation of non-zero principal ideals. The term "epimorph" w i l l henceforth mean "algebra epimorph," We retrieve the symbol R for a radical class of Boolean algebras. We make the convention that an ideal I of an algebra A w i l l be called an R-ideal of A i f A x e R for every x e I, Hence an R-ideal need not be in R, The following propositions summarize the properties of radical and semi-simple classes which we w i l l find most useful. Some of these properties have already been proved, and the others are straight-forward extensions of known results, •*-»27 Proposition; Let R be a radical class of Boolean algebras. Then: i ) 2 e Rp i i ) R i s hereditary, i i i ) i f A is an algebra with an R-ideal I such that A/I i s in R, then A i s in R, iv) 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 of weak products, 1,28 Proposition: Let M be a semi-simple class of Boolean algebras. Then: 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 in W, then A i s in M, CHAPTER TWO THE LOWER RADICAL Given any class of rings, i t i s possible to construct a smallest radical containing this class (see Divinsky [8]). We Investigate the special features of the general construction in the case of Boolean algebra radicals, §1, The Lower Radical Construction 2,1 PH^Jgosl^ion. Let X be any class of rings in some universal class of rings. For any ordinal a, we define a class X^ as follows: i ) x o - X, i i ) X^ i s the class of epimorphs of X-rings, i i i ) for a > 1, assuming X^ has been defined for a l l 3 < a, let X a be the class of a l l rings A such that every non-zero epimorph of A has a non-zero ideal belonging to X^ for some B < a. Let L(X) be the union of the classes X a„ Then L(X) i s a radical class. Furthermore, i f R is a radical class containing X, then R contains L(X)„ If H i s a class of rings closed under epimorphs and R = L(H), we have a construction due to Amitsur [2] which, for any ring A, yields an ideal h*(A) of A. Whenever H satisfies an additional condition, h*(A) coincides with r (A) , thus giving an internal. Iterative construc-tion of the radical of A in terms of H, 2.2 Definition (Pierce): A ladder in a ring A is a well-ordered chain of ideals of A, 0 <= I Q « 1^  $ ... < * a ^ » s u c n that i f a i s a limit ordinal, then I i s the union of the In for 3 < a. We " a 3 note that there is a least ordinal 6 such that I = Is for a l l a £ 6, a o * and we c a l l Ig the summit of the ladder { l a K 2.3 Lemma: Let R be a radical class and suppose {I aJ i s a ladder in a ring A with summit Ir, satisfying I ,,/! e R for a l l ordinals a, o" . a+1 a Then 1& i s in R. "Proof: ' UsingProp. 1.6 and'Prop, 1.8, an easy induction shows that I Q i s in R for a l l ordinals a, 2.4 Definition: Let H be a class of rings closed under epiroorphs. For any ring A, we define a ladder i n A as follows: i ) h Q(A) - 0 . i i ) assuming that h Q(A) has been defined, h^ + 1(A)/h (A) i s the ideal of A/ha(A) generated by Its H-ideals, and i i i ) i f a i s a limit ordinal and hg(A) has been defined for a l l 6 < a, then h o(A) is the union of the hg(A) for 3 < a. We c a l l {1^(A)} the tf-ladder in A and denote i t s summit by h*(A). Using Prop. 1.6S we easily get: 2.5 Lemma: If H i s closed under epimorphs and R » L(H), then for a l l rings A, h*(A) « r(A). 2.6 Def inition: A class of rings tf, closed under epimorphs, i s called an Amitsur class i f h*(A) = 0 implies h*(I) = 0 for 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) for a l l rings A i f and only i f H i s an Amitsur class. The proof of the sufficiency i s due to Amitsur [2], The necessity follows from the fact that semi-simple classes are hereditary. §2, The Lower Radical for Eoolean Algebra Radicals If X i s a class of Boolean algebras, we can use i t to gener-ate a Boolean ring radical, and thence a radical class of algebras L(X) which i s minimal with respect to containing X, This assures us that there i s a lower radical construction for Eoolean algebra radicals. Using the fact that ideals of ideals of an algebra are them-selves ideals of that algebra, we can show that the general construction of Prop, 2,1 stops at the second stage for Boolean algebra radicals. 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 ideal (necessarily non-zero principal) which i s an epimorph of an X-algebra. Then R » JL(X), Proof: Since X S R GL(X), i t suffices to show that R i s a radical class. Clearly, R i s closed under epimorphs. Now suppose that A i s an algebra such that every epimorph of A has an ideal i n R, We must . show that A i s in R, Suppose B i s an epimorph of A, By assumption, B has an ideal C in R, Then C, being an epimorph of i t s e l f and i n R, contains an ideal D which i s an epimorph of an X-algebra, But then D i s an ideal of B, and so every epimorph B of A contains an ideal which i s an epimorph of an X-algebra. Thus A i s in R as required, and R i s a radical cTas's, The following definitions and lemma w i l l be useful i n deter-mining the structure of algebras in a lower radical, particularly when the generating class i s closed under epimorphs9 and they w i l l also be extremely useful i n Chapter Five. 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 disjointed subset of an algeb-ra A with sup D ra 1, then A has a dense ideal-generated subalgebra iso-morphic to wH(A^: d e D), When A i s complete, we have A - lUA^: d e D), We extend these results 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 s a P-algebra. The property P i s hereditary i f , whenever x 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, for some hereditary property P$ then any maximal disjoint subset D of P-elements must satisfy sup D ™ 1, Hence, we easily get the following: 2»1Q Lemma: 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 in A, " Noting that the property of being a two-element algebra i s hereditary, we immediately deduce the following well-known result: 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 is 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 ideal i n H, i i ) A i s in 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, iv) any. complete algebra A in R is 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 08, Since principal ideal of an algebra are epimorphs of that algebra, i i ) follow from i ) . Using Lemma 2,10, we immediately get i i i ) and iv) from i i ) . and v) follows from i v ) , 2.13 Proposition; Let R = L(H) as in Prop. 2.12, and l e t A be a homo-geneous algebra* Then A is in R i f and only i f A is in H, Proof; If A i s in R, then every epimorph of A and in particular, A i t s e l f , has an ideal i n tf. Since A i s isomorphic to any principal ideal of i t s e l f , A i s in K, The other direction is obvious. Homogeneous algebras are a special case of the following more general concept: 2.14 Definition (Divinsky): A Boolean algebra A i s unequivocal i f for every radical class R, A i s either i n R or i s R-semi~simple, 2015 Proposition: An algebra A i s unequivocal i f and only i f L (AJJ) = L (A) for every non-zero x in A, Proof; We always have L(Ax) contained in L(A), If A i s unequivocal, then i t must be in L(A^) since i t cannot be semi-simple with respect to this radical. Hence we get L(A) = L(A ) for any non-zero x i n A, Conversely, suppose this always holds for an algebra A, and suppose that A i s not R-semi-simple, for some radical R» Then for some non-zero x in Ap A^ i s in R. But then, L (A) = *-( A x) i s contained i n R, and so A must be i n R. Hence A i s unequivocal. 2.16 Corollary: Every non-zero principal ideal 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 pairs exist: take A, = F and A„ «= P / I , where I i s the ideal of f i n i t e sets in P . i o 2 o " o Clearly, in any such situation, L (A) = L(A^) = 1 ^ 2 ) . Now let x = (x ±, X£) be a non-zero element of A, so that either x^ or x^ Is non-zero. Then A has a principal ideal, hence an epimorph, isomorphic to either A^ or A£. So 1(A X) = 1(A) for a l l non-zero x in A, and A i s unequivocal. We now show that the Aoitsur procedure for obtaining the radical i s always valid for Boolean algebras. 2.18 Proposition: If H i s a class of Eoolean algebras closed under epimorphs, then H is an Amitsur class. Proof: Suppose that A i s an algebra for which h*(A) = 0 and that I i s an ideal 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 class by Defn, 2,6, §3, The Superatomic Radical 2.19 Definition (Mostowskl and Tarski [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 f i r s t .->;S£ud±edi\by .Kostcwek^^ Day [6], [7], The following proposition from Day [7] summarizes the various characterizations of superatomic Boolean algebras, 2.20 Proposition: If A is a Boolean algebra, 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, iv) 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 rational numbers, v i i i ) the Stone space of A is clairseme; that i s , every non-empty subspace of S(A) has an isolated point. If we let H be the class consisting of the two-element algebra, then H i s closed under epimorphs, and Defn. 2.19 can clearly be re-stated as follows: A i s superatoiaic i f and only i f every epi-morph of A has an ideal in ff. By Theorem 2.12, we have: 2,21 Proposition: The class 0 of superatomic Boolean algebras i s a radical class; namely, i t i s L(2), the lower radical generated by the two-element Boolean algebra, 2o*22 Corollary: Every radical class of Boolean algebras contains 0, Proof: Every radical class of Boolean algebras contains J2. 2.23 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 radical of any algebra. The following i s clear: 2.24 Proposition: Let H =» {2} and l e t A be any Eoolean algebra. Then h ^ (A)/h^(A) i s the Ideal of A/h o(A) generated by i t s atoms; more precisely, i t consists of a l l f i n i t e joins 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 least ordinal for which h.(A) = A, 6 2.25 Definition (Day [7}): The cardinal sequence of the superatomic Boolean algebra A i s the sequence of order type 6 whose a-term, for a < 6, i s the cardinality of the set of atoms of A/h a(A), This specialization of the Amitsur construction has been used with success by Day in his study of superatomic algebras. We mention two of his more striking results. 2.26 Proposition (Day [7]): Two countable superatomic Boolean alg-ebras are isomorphic i f and only i f they have the same cardinal se-quence, 2.27 Proposition (Day [7]): If k i s an i n f i n i t e cardinal, then there are more than K non-isomorphic superatomic Boolean algebras of cardin-a l i t y K„" Superatomic Boolean algebras have arisen naturally in the study of free complete extensions of an algebra. 2.28 Definition: C is 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 ex-tended to a complete homomorphism (that i s , one which preserves a l l suprema) of C to that algebra. If B has a free complete extension, then i t i s unique up to isomorphism, 2.29 Proposition (Yaqub [27], Day [6]): B has a free complete extension i f and only i f B i s superatomic. Further results of a similar nature occur i n these papers. The Cardinality Radicals 2.30 Definition: For any ordinal a, we c a l l the lower radical generated by F a, the free algebra on Ha generators, a cardinality radical and denote i t by F « 2.31 Proposition; F^ i s the lower radical generated by the class of Boolean algebras of cardinality at most H » Moreover „ i f o < a then F A i s properly contained in F^ . Proof; The f i r s t assertion follows from the fact that every algebra of cardinality at most H i s an epimorph of F , If o < &, F i s an a a * a epimorph of Fg S so we get F^ contained i n F^, But F^ i s in F^ and not in F A „ so the containment i s proper. In a natural sense B the superatomic radical i s the f i r s t member of this chain, for i t i s the. lower radical generated by a l l f i n i t e Boolean algebras. We shall see in Chapter Seven that F q i s an atom i n the la t t i c e of Boolean algebra radicals, so we have some interest i n examples and some properties of F o-algebras. 2,32 Examples; Of course, any weak product of countable algebras is i n F . Furthermore, any atomless algebra i n F Q has a dense ideal-generated subalgebra 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 realized; 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 in P(tc), which can be realized as the subalgebra of elements with {0, l}-coordinates in any K-product. In particular, we can assume that P(K) i s embedded in F q , Let I be the ideal of F Q generated by the atoms of S, Then any non-zero principal ideal of F Q contained in I i s isomorphic to F Q . Furthermore, i f A i s the sub-algebra of F^ generated by S and I, then I i s also an ideal of A, and A/I i s superatomic. In fact, A/I is isomorphic to the quotient of S by the ideal generated by i t s atoms. Hence, by Prop. 1.6 and Prop, 2,22, A is in F q , Intuitively, we can regard this as the replacement of each atom in S by a copy of F Q . In Chapter Six, we w i l l describe another method of obtain-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 inition: K i s the lower radical generated by the separable Boolean algebras. We w i l l find use for this radical in Chapter Seven. 55. The Power-Set Radicals 2.34 Definition: The lower radical generated by P a, the algebra of a l l subsets of a set of c a r d i n a l i t y # a , w i l l be called a power-set  radical, and denoted by P .. 2.35 Proposition: Let HD - exp ft . Then P - L(F„). -— 3 'a a (3 Proof: By Prop. 0.8, P Q and Fg are epimorphs of one another. 2.36 Proposition: If a ^ 6, then ? a i s contained in Pg. Further-more, ?a - Pg i f and only i f exp fta a exp ftg. Proof: The f i r s t assertion follows from the fact that P i s an epi-morph of Pg, and the second from the equivalence of the cardinality condition with the fact that P Q and Pg are epimorphs of one another, using Prop. 0.9. 2.37 Theorem: Let ft « exp ftQ, Let A be a complete algebra. Then the following are equivalent: 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) |A| Proofs Using Prop, 0C5 and Prop, 0.8 0 we see that i i ) e i i i ) and iv) are equivalent, and they clearly imply i)» Now assume i ) . Then by Theorem 2.12, A i s a product of epinorphs of P Q, say A = H(A^; i e 1). Let 0 H Q » T h e n Pg can be represented as & product of |l|-copies of P a„ each of which has one of the A^ as an epimorph. Then Pg has their product A as an epimorph, We must now show that Pg i s an epi-morph of P . This i s obvious i f 8 < a so assume a < 8. Since HD m a p max (Hat this means that = Since A is in PQ, and has KP(1) »"Fg as a retract, this iiieans that "is in P0»"hence that « P a. .. But this i s equivalent to the fact that P a and Pg are epi-morphs of one another. 2.38 Corollary; Let A be an i n f i n i t e complete algebra. Then A i s in ? o i f and only i f |A| « exp HQ* In this case, L(A) » P Q and i f A is atomless, then i t is unequivocal. Proof; Any i n f i n i t e complete algebra has cardinality at least exp $iQ, The rest follows from the Corollary to Prop. 0.8 and from Prop. 2,15. It i s interesting to note, here, that Monk and Solovay [20] have shown that there are exp exp Ha- isomorphism classes of complete Boolean algebras of cardinality exp Ha. We w i l l be interested i n Chapter Five with those complete algebras that generate a power-set radical, 2.39 Theorem; Let = exp f ^ . Let A be any algebra i n ? a. Then the following are equivalent; i ) L(A) - P A P i i ) P a i s an epimorph of A, i i i ) Fg i s an epimorph of A, iv) Fg i s a subalgebra of A. If A i s any complete algebra, then A generates P q for i t s lower radical precisly when i t has cardinality and satisfies any one of i i ) , i i i ) or i v ) , which is equivalent to saying that A and ? a are epi-morphs of one another. Proof; Using Prop, 0.6 and Prop. 0.6, we see that i i ) , i i i ) and iv) are equivalent to one another, and i t i s clear that they imply i ) . Assuming i ) , we see that F^ is in L (A), Being homogeneous, i t i s an epimorph of A by Prop. 2.13. Thus i) implies i i i ) . If A i s a complete algebra, then A e ? a i s equivalent to the inequality |A| S W^ e which, i n the presence of any of the conditions i ) - i v ) , must actually be an equality. The rest follows immediately. Finally, we introduce a related radical; 2.40 Definition; P i s the lower radical generated by the class of a l l complete algebras. 2.41 Proposition; P i s the lower set algebras, and so is the smalle the P O.. Proof: Every complete algebra i s radical generated by a l l power-st radical class containing a l l a retract of a power-set algebra CHAPTER THREE THE UPPER RADICAL Given any class of rings satisfying a condition called regularity, i t i s possible to generate a semi-simple class which is minimal with respect to containing the given class,, §1. The Uffjagr Radical Construction We describe the construction immediately for classes of Boolean algebras 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 recall that the two-element algebra must be in any radical class. This entails that semi-simple algebras must always be atomless, 3,1 Definition: A class H of Boolean algebras w i l l be called 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 class. 3.2 Proposition: Let M be a regular class of Boolean algebras, and le 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 radical class of algebras, i i ) every algebra in W i s semi-simple with respect to U(M), i i i ) i f R i s a radical class such that every algebra in M i s R-semi-simple, then R i s contained in U(M). 3.3 Definition: The radical class U(M) is called the upper radical determined by the class W, Given any class V of atomless Boolean algebras, i t i s clear that the class U of a l l non-zero principal ideals of y-algebras is a ,her.e,d itar-ycla s s. ...o f „ .atomle ss, algebras.. 3.4 Definition: If y i s any class of atomless Boolean algebras and M i s the class of a l l non-zero principal ideals of ^-algebras, then we w i l l denote the radical class ti(M) by U(/) and c a l l i t the upper radical determined by the class V» §2. The Characterization Theorem 3.5 Theorem: Let M be a regular class of Boolean algebras, and l e t U(M) be the upper radical determined by M. Then an algebra A i s U(M)-semi-simple if 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 direction i s immediate. The other direction w i l l follow i f we can show that for R » U(W)« and for any Boolean algebra A, r(A) i s equal to the intersection J of a l l ideals I of A such that A/I i s in W. Then, for a semi-simple algebra, this intersection 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, let x be an element of J which is not i n r (A). Then A^ i s not i n R, and so has an epimorph A/K i n Note that K i s also an ideal of A. Let L denote the ideal of A generated by K. and x*. Then, using §2, f) of the Preliminaries, we see that A/L - A^/K, which i s in 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 in L and, being disjoint from x', i t must be in K, But this yields the contradiction A^/K = 0. Hence J » r(A) as required, ^ •' Atomless Boolean Algebras The superatomic radical 0 Is clearly the upper radical deter-mined by the atomless Boolean algebras. We find a much smaller regular class which determines 0 as i t s upper radical. 3«6 Theorem; Let M be the class of a l l atomless, separable Boolean algebras. Then: i ) M i s a hereditary class, and so determines an upper radical, i i ) this radical i s , in fact, the superatomic radical, 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 clear 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 q ; that i s , any non-superatomic 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 clear that a class consisting of a single atomless, homogeneous algebra i s a hereditary class, and so generates an upper radical. Any semi-simple algebra, then, can be represented as a sub-direct power of this algebra. The radicals defined in this section a l l have this feature. In only one case do we make further'mention of this fact, for i t yields a subdirect power representation for atomless, complete algebras. 3.7 Definition; E a i s the upper radical determined by the i n f i n i t e free algebra F a, 3.8 Proposition; If a < B, then £ o is properly contained i n E^. Proof: The containment follows easily from the fact that F i s an epimorph of Fg. Furthermore, F a i s in E^ but not in E a, 3.9 Definition; 6^ i s the upper radical determined by F a, 3.10 Proposition; If a $ 8, then G Q i s contained in Gg. Furthermore, G « GQ i f and only i f F and F 0 are epimorphs of one another, U P a 8 Proof t F^ i s an epimorph of Fg. The rest i s obvious. 3,. 11 , Corollary: Let K, « exp.£L .and l e t o a,,(3 .« jy. Then G„ » Gp. Proof; By the Corollary to Prop. 0,8, any two complete algebras of cardinality exp HQ are epimorphs of one another. For F and F to be mutual epimorphs, i t i s necessary that ft 8 their cardinalities be the same: Ha 0 ** °<> We do not know whether the cardinality condition i s sufficient. 3.12 Theorem; Any atomless, complete algebra i s a sudirect power o f F D . Proof; Using Theorem 3.5, i t suffices to show that any atomless, complete algebra i s G^-setai-simple, Any principal ideal of an atom-less, complete algebra i s i t s e l f atomless and complete, and so, by §3, j ) of the Preliminaries, i t must have F Q as an epimorph and cannot be in G q C Hence any atomless, complete algebra i s G0-semi-simple. We shall see that GQ has many upper radical characterizations and so yields many subdirect product representations for atomless, com-plete algebras. Because of i t s special properties, however, F Q i s an especially appropriate building 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 unit interval [0, 1) of the reals. It 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 unit interval [0, 1] of reals by s p l i t t i n g every interior point x into two parts, — + x and x , We consider X as an ordered set with the natural order: — + + 0 < x < x < y~ < y T < 1 whenever 0 < x < y < 1, and give i t the order topology (see Sikorski [24], example §9, E), We w i l l show that Q i s i n both G Q and E C , If Q i s not i n GQ, then i t has P Q as an epimorph; i n other words, we can embed gN i n X, But |x| «= exp HQ and j gN j « exp exp Hq9 so this i s impossible, ' ' Ho If Q i s not i n fcQ, then we can embed the Cantor set 2 i n X, To show that this i s impossible, we show that any uncountable closed subspace of X has an uncountable base. Let F be any uncountable closed subspace of X and l e t {Ga> be a base for F. Let F + (respectively F~) be the points of F of the form x + (respectively x~) , Then one of F +, F~ must be un-countable. Suppose F + i s uncountable. For any x + in F +, the inter-val [ x + s 1] i s clopen i n X and so F O [x +, 1] i s clopen i n F. Thus there i s a basic set G x such that x + e G^Q F f l [x+, 1], If x + < y +, then x + i [y +, 1] and so x + i Gy. Thus for distinct x +, y + in F +, we get distinct basic open sets G x and Gy, and the base {GQ} must be uncountable. If F i s uncountable, an obvious modification of the argument yields the same result. §5, The Upper Radical Determined bjr Homogeneous Algebras 3.14 Definition: J i s the upper radical determined by a l l atomless, homogeneous algebras. J i s contained i n any upper radical determined by a class of atomless, homogeneous algebras. Thus i t i s contained i n both GQ and E Q . It i s not equal to their intersections however, for the algebra Q of the l a s t example i s J-semi-simple, A natural question here i s whether J = (?, In Chapter Seven, we w i l l discuss one consequence of a positive answer. We present some considerations which make a positive answer reasonable. 3.15 Definition; A monotonic cardinal property v assigns a car-dinal number v(A) to each algebra A in such a way that v(A x) $ v(A) for a l l non-zero x in A. If this inequality i s an equality for a l l non-zero x In A, then A i s called 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 in A. Proof: For any non-zero x in A, pick y £ x such that v(A^) i s minimal among the v(A g) for 0 $ z < x. By monotonicity of v, A y i s v- homo-geneous. 3.17 Corollary: If v i s a monotonic cardinal property, then any complete algebra i s a product of v~homogeneous algebras. Proof: It i s clear that the property of being v-homogeneous is her-editary, so we can apply Lemma 2,10, This last result i s due to Pierce [22] and provides support for his conjecture [23] that every complete algebra i s a product of homogeneous algebras. We provide similar support for the conjecture that J * 0e 3.18 Proposition: Let v be a monotonic cardinal property and l e t J v be the upper radical determined by the class of atomless, v-homo-geneous algebras. Then J v = 0a Proof % Any non-superatomic algebra has an atomless epimorph, which has an atomless, v-homogeneous principal ideal. Hence J v can contain only superatomic algebras. CHAPTER FOUR CRAMER*S RADICALS We investigate the radicals introduced by Cramer in [5], §1.The Classes Superatomic Boolean algebras can be characterized as those having no countable free subalgebra. Cramer generalized this as follows: ^•^ Definition: The class C Q i s the class of a l l Boolean algebras with no subalgebra isomorphic to Ffl, Cramer's proof that the C Q are radical classes uses topolo-gical methods. We present an algebraic proof. 4.2 Proposition; Ca i s a radical class. Proof: By projectivity, whenever F a can be embedded in an epimorph of A, then F a can be embedded i n A, 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, If F a can be embedded in A, then by Prop. 0.1, F a can be densely embedded in 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 principal ideal of F Q generated by y can be embedded i n which can be embedded as a subalgebra in Bx, Since i s homogeneous, this says that no non-zero principal ideal of B can be i n C a, contra-dicting the assumption on A. Thus A is in C a, and by Prop, 1,21, C i s a radical class, a 4.3 Proposition; If a < 6, then C a i s properly contained i n Cg. Proof: The containment follows from the fact that F a can be embedded in Fg. It i s proper since F a e Cg but F Q i C a. §2 . The C a as Upper Radicals 4.4 Definition: The class X>a is the class of a l l ,Boolean,,algebras which do not have P a as an epimorph, 4.5 Proposition (Cramer [ 5 ] ) : Let Hg ra exp H^, Then VA •= Cg. Proof: This follows immediately from Prop, 0,6, 4.6 Corollary: VA Is a radical class, and i f a < 8, then P q i s properly contained i n Pg, Except for the fact that P a i s not atomless, the description of P Q suggests an upper radical. If a exp HaS however, since P a and Fg are epimorphs of one another, we immediately get Cg => P Q •= Gg. Actually, we can extend this result. 4.6 Proposition; i s contained in G q 9 and C Q ° G^ i f and only if Proof; Since Fffi has F a as a subalgebra, i t i s not i n C a and being homogeneous, i t must be C a-serai-siiaple. But G A i s the largest radical for which Fft i s semi-simple, so C Q i s contained in G^t If Ha ° "Hat then |Fa | «= and so i t i s an epimorph of F a„ If F Q i s embedded in an algebra A, then by i n j e c t i v i t y of F Q, we get that A has F Q as an epimorph. In other words, Ga i s contained in C Q, and we get equality of the radicals. Conversely, suppose the radicals are equal, and suppose that ^ a 0 > //a° By cardinality, then, F a cannot have F Q as an epimorph and so F a i s in G A, But F a £ C Q, so this contradicts the equality of the radical classes. §3« The Radical VQ Since P and F are epimorphs of one another, VN RA G . We O O r r " O O have already seen that a l l atomless, complete algebras are ^-semi-simple. This section plays variations on the theme that P Q-algebra8 are in a very strong sense the opposite of complete algebras. The basic fact we need i s Pierce's result (53, j ) of Preliminaries) that any i n f i n i t e epimorph of a c-complete algebra must have P Q as an epi-morph. An immediate consequence of this i s the following: 4.7 Proposition; Let A be an i n f i n i t e P Q-algebra. 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 class of atomless algebras satisfying: i) each algebra i n V has P q as an epimorph, and i i ) there i s an algebra in V which i s an epimorph of P Q . Then the upper radical determined by V i s P Q , Proof; By condition i ) , a P Q-algebra cannot have an epimorph in Yt so VQ i s contained i n the upper radical determined by V, By con-dition i i ) , any algebra in the upper radical determined by V cannot "have"P0 as an epimorph, and so i s *ih* P Q . 4.9 Corollary: VQ i s the upper radical determined by any of the following classes; 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, iv) the class consisting of a l l principal ideals of P A / I » where I i s the ideal of f i n i t e sets i n P a, v) the class consisting of F Q , vi) the class consisting of any atomless, complete algebra of cardinality exp h{ . CHAPTER FIVE DECOMPOSITIONS OF COMPLETE ALGEBRAS The product decompositions of this chapter depend on fi n d -ing a dense subset of P-elements in an algebra, for some hereditary property P. The search for P-elements below an arbitrary non-zero element leads naturally to the consideration of various chain-like conditions. §1. The General Setting We have already seen that Pierce's decomposition of complete algebras v i a cardinal properties i s a special case of Lemma 2.10. The theorems of this chapter also make use of this lemma. We note that the properties of being homogeneous and unequivocal are hereditary prop-erties. Hence, we immediately get the following: 5.1 Proposition: Let A be a complete algebra. Then A is a product of homogeneous (unequivocal) algebras i f and only i f the homogeneous (unequivocal) elements of A are dense in A. Pierce's result also included a uniqueness feature (see[22J) which also holds for decompositions into homogeneous (unequivocal) algebras, whenever such decompositions exist. The following propo-si t i o n includes a l l these uniqueness results as special 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 relation = 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 follow-ing properties: i ) for any x e X, A x i s a product of P-algebras A y, y E Y X, where y-^  = yg for any y^, y2 e Y^p and i i ) for x i z in X, e Y^, y 2 e Y t y^ t y2* Proof: The set X consists of the suprema of the equivalence classes of P-elements of A, Then X i s disjointed by (*), and sup X = 1 by the density of P-elements in A. The rest of the proof i s a straight-for-ward ve r i f i c a t i o n . To see how this applies, v/e need to specify an equivalence relation for 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 ) . x y In each case, i t is easy to see that condition (*) i s satisfied, so that a suitable replacement of P in Prop, 5,2 w i l l yield a uniqueness result in each of these three situations, §2, Decompositions into Homogeneous Algebras 5.3 Definition; An algebra A w i l l be called near-homogeneous i f every descending chain of principal ideals of A contains only a f i n i t e number of isomorphism types of algebras. Finite products of homogeneous algebras are a natural example of near-homogeneous algebras. Another class of examples are the power-set algebras. Any principal ideal of such an algebra i s another power-set algebra, which i s determined up to isomorphism by the cardinality 6f i t s atoms. Hence any descending chain of such ideals yields a descending chain of cardinals B which must be f i n i t e . 5.4 Theorem; If A i s complete and near-homogeneous, then A is a pro-duct of homogeneous algebras. Proof: By Prop. 5.1, i t suffices to show that any non-zero principal ideal of A contains a non-zero homogeneous principal ideal. So l e t x be a non-zero element of A, If A i s not homogeneous, there i s a non-zero element y 5 x such that A i s not isomorphic to A . Proceed-y x ing inductively, we get a descending chain A >A >,.,> A > ... x y z where no two adjacent algebras are isomorphic. By §3, g) of the Preliminaries, i f any two algebras in the chain are Isomorphic, then they are also isomorphic to a l l the intervening ones. Hence no two algebras in the chain can be isomorphic. Because A i s near-homogeneous, the chain must terminate in a f i n i t e number of steps, and the algebra thus obtained i s a non-zero homogeneous principal ideal of A contained in 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 K n s n < w} be a s t r i c t l y increasing sequence of cardinals satisfying K n ° " ^n* Such sequences exist: for example, take KQ = exp //o and K n + 1 = exp For any such sequence, l e t A n denote the normal completion of the free algebra on tcn generators. Then A Q Is complete, homogeneous, and i t s cardinality i s tc^. Let A *» H(An: n < u). There i s a natural isomorphism between certain principal ideals B^ of A and par t i a l products of the A^ as follows: B^ a E(A n: k « n < w). Thus we get a descending chain of ideals of A: B Q > B^ > ... > B^ > Any non-zero principal ideal of B^ must contain a copy of some A Q for n £ i and so must have cardinality at least For j < i , howevery, Bj has a principal ideal isomorphic to Aj of cardinality Kj. Since Kj < K^, B^ cannot be isomorphic to Bj. §3. Decompo s i t ion s into Unequivocal Algebras 5.6 Definition: Let {Ka> be a well-ordered chain of radical classes. We say that an algebra A i s R-layered i f , for any non-zero x in A, either A i s f i n i t e , or there i s an ordinal 0 such that L(A ) « R0. A X P A w i l l be called layered i f there exists some well-ordered chain of radical classes {Ka> such that A is R-layered, Examples of layered algebras w i l l be given in §4. It i s clear that any principal ideal of an R-layered algebra i s i t s e l f R-layered, 5.7 Theorem: If A i s a complete layered algebra, then A is a pro-duct of unequivocal algebras. Proof: By Prop. 5.1, i t suffices to show that any non-zero principal ideal of A contains a non-zero unequivocal principal ideal. So l e t x be a non-zero element of A, and suppose {&a) i s the chain of radicals 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 least a such that I (Ay) «• R q for some non-zero y .$ x. Choose some y for which l(Ay) » R^ , Now for 0 ^  z •$ y, i ( A z ) i s contained in Rg, Since A i s R-layered, 1(A Z) i s some Ra, and by the minimality of &, L(A Z) <= Rg, But then, by Prop, 2,15, A i s an unequivocal principal ideal of A^. 5.8 Theorem; Let A be a complete algebra, and suppose that for any non-zero x in 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 is precisely what i s needed to make A a P-layered algebra for the chain Using Theorem 2,39, we see that i s would be extremely useful, i n determining the scope of this theorem, to know which algebras, other than power-set algebras and completions of free algebras, have large free subalgebras. Unfortunately, l i t t l e i s known. As a sample, we quote the following.result: 5.9 Proposition (Efimov [11]); For any algebra A, let cA denote the supremum of the cardinalities of families of disjoint 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 in the lower radical generated by an i n f i n i t e complete algebra. Hence there i s no point i n attempting further results along these lines using the chains {F Q}f o r ' §4, Connections with Cardinal Properties It i s possible to obtain Theorem 5,7 in a slightly more lengthy manner using Pierce's result (Cor, 3.17). There are some interesting additional results 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 radical classes {Ra}, we say an algebra A i s admissible with respect to the chain i f i t is i n one of the radicals of the chain. For any admissible A, we define p(A) = min {Hai A e R }. Then p i s a cardinal property on admissible algebras, and the fact that i t i s monotonic follows easily from the fact that every radical class i s hereditary. We note that p(A) is always i n f i n i t e . 5.11 Lemma; The admissible algebra A i s p-homogeneous i f and only i f there i s an ordinal 6 such that: i ) A e &a for a l l a ^ g, i i ) A is R -semi-simple for a l l a < 6. a In this case, of course, p(A) = H^-Proof: Let A be p-horaogeneous with p (A) » ft e Then for a l l a J. 6, b A i s in R . By p-homogeneity, p(A) = p(A) = ft for a l l non-zero x o x e in A. In other words, no non-zero principal ideal of A occurs i n any R a for a < £. Hence A i s Ra-semi-simple for a l l a < 6. Con-versely, suppose such a g exists. Then clearly p(A) « H^, Since A is R -semi-simple for a < 8, A cannot be in any such R for any nOn-Ct X Cl zero x i n A. Since p(A x) p (A) •=• K g , we must have p (A^) " /fg« Hence A i s p-homogeneous. 5.12 Lemma; Let {RQ} be a well-ordered chain of radicals. Suppose the algebra A i s R-layered (hence admissible) and p-homogeneous. Then A i s unequivocal. Proof; Let p (A) =» //ot) and let x be a non-zero element of A. By 1 1 1 • P p-homogeneity, A cannot be in R for any a < 8. Hence, for any such x " q, • Lj(A •) f .(?a«'. Since A i s R-layered and_i-,(A ) i s contained in Rg, we must have L (A ) => R „ for any non-zero x i n A, Hence A i s unequivocal, x p 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. By Cor. 3.17, A is a product of p-homogeneous algebras. Being principal ideals of A, these algebras are also R-layered and so each i s , i n fact, unequivocal. We are now ready to proceed with examples. We concentrate on the chains {Pa} and {Pa> which define cardinal properties u and 6 respectively. Note that every complete algebra i s admissible with respect to these chains. For the remainder of this chapter, we assume GCli. Aside from the fact that GCH simplifies the examples we consider, the following proposition requires the assumption that i f a < 6, then exp H < exp # . a p 5.14 Proposition (GCH): For any admissible algebra A, 6 (A) < ir(A)+. Furthermore, 6(A) s ir(A)+ i f and only i f L(A) = P^ where #a » if (A). Proof: Since P a cannot have P a + i a s* a n epimorph, we get P Q C. #A^» This implies the f i r s t statement. Clearly, <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 as an epimorph. By Theorem 2.39, this i s equivalent to L(A) » P , 5.15 Corollary (GCH): An algebra A i s P-layered i f and only i f , for every non-zero x in 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 this property is a product of unequivocal algebras. 5.16 Examples (GCH): We note that ir(P a) = Ha and 6(P Q) =» %ot+l« Hence every power-set algebra i s P-layered. Since P^ *° L ( F a + ^ ) , we have * ( F A + 1 ) 8 3 h'a» Now suppose a i s a l i m i t ordinal. We always have that'-F i s an epimorph of P a, so F a E P a, Suppose B < a and F a e Pg Then F Q i s an epimorph of Pg V i so } fta °, But 6 < o implies B + 1 < a and K+, .< Ha °» So ¥ a i s not in Pg, 0 < a, and ir(F 0) » a . In either case, T K F Q ) ° £(Ky: y < a). Since F a + i has P a as an epimorph, then, by i n j e c t i v i t y of P Q, so does F 0 + i . Hence 6 (^a+A^ "^a+l*' ^ o r s u c c e s s o r ordinals, Fa+1 i s P ~ l a y e r e d » For limit ordinals, the situation i s unclear. For example, we do not know i f F i s P-layered. However, i f a i s a limit ordinal H — — satisfying Hr, ° : S 8 „» then IF I " H„t so F cannot have P as an epi-morph; that i s , F a e Va. If 6 < a, 'then 3 + 1 < o and so F Q has Pg as an epimorph. So then does F Q, and F a t for 8 < a. In this case, then, <5(F ) = Hn « f(F„), Hence, i n this case, F„ i s not P-layered. We note that such ordinals exist; for 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 radical class i s closed under f i n i t e products and weak products. Every semi-simple class i s closed under subdirect products. We extend these results. §1. Closure of Radical Classes under Products Any K a-product of algebras has P as an epimorph,, so this gives us a crude negative result: whenever a radical does not contain P a o then n o - p r o d u c t can be i n the radical. One might hope to show the converse: whenever P i s in a radical class, then i t i s closed under ^-products. Cramer [5] has obtained a result which shows that this i s fal s e . 6.1 Proposition: For any ordinal a, there i s a sequence i^n' n < w} of superatomic algebras whose product i s not in C a« We shall see In Chapter Seven that any ? a is contained, i n some Cg, so we can find a sequence of superatomic algebras whose pro-duct i s not i n P Q, In Chapter Seven, we w i l l present a weaker form of this (false) conjecture which has more likelihood of being true. 6.2 Proposition: If exp fta ra exp ft t then P^ i s closed under ft products of complete P a-algebras. Proof; Let A = Il(A^: i e I) be a product of complete P Q-algebras with | l | « H$« By Theorem 2.37, |A±| « exp fta and so |A| <S (exp # 0/^ -exp fta» Since this product i s complete, i t i s in ? a by Theorem 2,37. Halmos ([14], exercise 3, p, 118) has defined a "weak pro-duct" slightly different from our weak product. We give a generaliza-tion of his 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 in 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 differ from an element of B in 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 ideal 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: It i s clear that C contains the Let x e C and suppose that x dif f e r s from elements b^, e B in at most a f i n i t e number of coordinates. Then b-^  d i f f e r s from b2 in 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 Proposition: Let R be a radical class and suppose the A^ and B are i n R . Then the product C of the A^ over B i s in R . Proof: J i s generated by radical ideals of C, and C/J, being isomor-phic to an epimorph of a radical algebra, i s radical. Thus C e R . 6.6 Corollary: Radical classes are closed under weak products. 52. Closure of Radical Classes under Coproducts 6»7 Easic Lemma: Let R be a radical class. Suppose A and B are Boolean algebras, and that A i s the summit of a ladder {1Q} with the following property: (*) for each a, each element of I a + j _ / I i s a f i n i t e j o i n of cosets [aJ such that (A/I ) ^ a j + B e R . 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 suffices to show that K a + A / ^ a e ^  £ ° r a l l <*, ^a+i/^a i s a n Ideal °f A + B/K^, which, by Prop e 0,4, i s isomorphic to A/I a + B0 As an ideal of A/I 0 + B„ the elements of K ^ ^ / i ^ can be represented as f i n i t e joins of elements of the form [a] A b where [a] e I a + i / l a and b e B, By (*)„ [a] » [a x] v ... v [a n] where ( A / I 0 ) [ a i ] + B e R, This latter algebra i s isomorphic to (A + B/K, ) r , which contains (A + B/K ) r . , , a laiJ . a [a^jAb Then, since each [a^l A b i s a radical element of Kg^/K » we get that K Q +j/K a e R as required, 6.8 Theorem: Let R be a radical class and let B e R. Then, for any superatomic algebra A, A + B e R, Proof: By Prop, 2,24, A is the summit of a ladder {l o} where every element of I a + i / I a i s a f i n i t e j o i n of atoms [pi e A/I^. Then ( A / l a ) [ p j + B«=2 + B= B e R . 6.9 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 less than or equal to ft, the f i r s t uncountable ordinal. Then the algebra S of clopen subsets of Z is superatomic, and so, by the l a s t theorem, A >• F-. + S e F , We show that i f I i s the ideal of o o A generated by the elements x such that A^ - F Q, then A/I Is not 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 B Now suppose M i s a clopen subset of X such that (y, &) e M for some y £ Y, The projection 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 z~*[Nj i s a clopen subset of X contained in M. Since Z has no countable base, neither, then, can M. Hence U n {(y, 0): y e Y} = <j>. Clearly, { (y, Q): y e Y} i s homeomorphic to Y, so X - U has a closed subspace homeomorphic to Y. Algebraically, A/I has F q as an epimorph, and so cannot be superatomic. 6.10 Lemma: Let H be a class of Boolean algebras closed under epi-morphs. Suppose B i s an algebra whose coproduct with any f/-algebra i s in JL(tf). Then the coproduct of B with any L(H)-algebra i s in L(H). Proof; 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 a + i / I a i s a f i n i t e join of H-elements. By the assumption on B, i t s coproduct with any principal ideal generated by an H-eletsent must be in L(tf). Hence, by Lemma 6.7, A + B e L (H) . 6.11 Theorem: Let X be any class of algebras closed under f i n i t e coproducts. Then L(X) i s closed under f i n i t e coproducts. Proof: 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 their co-product, 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 in L(H). Since this i s true for any C in H, we have that the coproduct of B with any H-algebra i s in L(H). Applying Lemma 6.10 again, we get that the coproduct of B with any L(H)-algebra A i s in L(H). Hence L(H) i s closed under f i n i t e coproducts. Since L (X) = LOO, the result follows, 6.12" Corollary; i s closed under f i n i t e coproducts. Proof: The class of algebras of cardinality at most Ha i s closed under f i n i t e (in fact, ^ a~) coproducts. One might hope to prove that f i s closed under H -coproducts. ct Vie w i l l show, in the next section, that i t i s not even closed under countable coproducts of superatomic algebra. The following result i s from Cramer [5]: 6.13 Proposition: Suppose that {A i: i e 1} is a collection of C Q~ algebras, f i n i t e coproducts of which are i n C . ; Suppose that | l | » £fg and that Ha i s //^-inaccessible (that i s , fta cannot be expressed as the sum of Hc cardinals each of which i s less than //„). Then the coproduct A •» Z (A.: i e I) i s in C . . l a Proof: The coproduct A is 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). Note that | j | « |X| « H . If A has a free subalgebra generated by a set D of cardinality Hai then since i s K^-inaccessible, D H must have cardinality hfa for some j e J. But D n Bj generates a free subalgebra of Bj, contradicting B^  e C^. 6.14 Corollary: Let {A^ i e 1} be a collection of F a-algebras. Suppose that | l | = H$ and that Ha+i i s /^-inaccessible. Then the co-product of the A^ i s in Proof: Since F^ does not have F + ^ as a subalgebra, F a i s contained in C a +^, and F^ i s closed under f i n i t e coproducts. Intuitively, this 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 projective, i t i s not unreasonable to ask i f f i n i t e coprod-ucts of complete algebras are complete. It i s relevant i n this con-text since a positive 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 Proposition: If A and B are i n f i n i t e algebras, then A + B i s not complete. Proof: Choose i n f i n i t e disjoint collections {a^: i < u} and {b^: i < «} in A and B respectively. Set x^ ° a^A b^ in A + B. Let x e A + B be an upper bound of {x^: i < u} . We show that x cannot be a least upper bound. We note that x can be represented i n the form x = (c^ A d-^ ) v .,. v ( c n A d n) where c^ e A and e B. Then there exist k and i £ j such that x^ A (c^ A d^) $ 0 and x j A (c^ A d^) ^ 0 S for otherwise, x would intersect only f i n i t e l y many x^. In other words, (a^ A CJ^ ) A (b^ A d^) ^ 0 and (a^ A c^) A (bj A d^) f 0. Then, since A and B are independent sub-algebras of A + B, we get that y » (a^ A C^) A (bj A d k) & 0. For any m < w, since i ^ j , either a.^  A a m = 0 or bj A b m » 0. Hence XJJJ A y <° 0 for a l l m < u. But 0 £ y •$ x so x A y' i s an upper bound of (x n: n < wl which i s s t r i c t l y smaller than x. This proposition generalizes Exercise 6N of Gillman and Jerison [12], where i t i s asserted that P a + P^ i s not complete. 6.16 Corollary: The coproduct of two algebras i s complete i f and only i f one i s f i n i t e and the other complete. Proof: This follows immediately from the proposition and the fact that i f A i s f i n i t e with n atoms, then A + B - B n. Using the fact that 0 i s closed under f i n i t e coproducts, we are able to prove the following: 6.17 Proposition: EQ i s closed under f i n i t e coproducts. Proof: Suppose that A and B do not have as an epimorph. If f i s an epimorphisra of A + B onto F with kernel K, l e t I » K f) A and J » K n B be the corresponding ideals in A and B. Let L be the ideal of A + B generated by I and J . Since L i s contained i n K, A + B/L has A + B/K - F Q as an epimorph. The eplmorphism f maps A onto the subalgebra A/I of F o« Hence A/I i s countable, so the only atomless epimorph i t could have is F Q. Because A e E Q, this cannot occur, so A/I is superatomic. Similarly, B/J i s superatomic, and so, then, i s their coproduct A/I + B/J. However, A/I + B/J - A + B/L, and we have already shown that this algebra has F Q as an epimorph. This contradiction shows that A + B .e E . o 6.18 Corollary; Let X be the product of the Boolean spaces X^, ... a X n. 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 results of this section indicate that coproducts are far more l i k e l y to be semi-simple than radical. 6.19 Theorem; Let R be a radical class and suppose the collection {A^: i e 1} contains at least one R-semi-simple algebra. Then the coproduct of the A^ i s R-semi-simple. Proof; Clearly, i t suffices to show that i f A is R-semi-simple, then so i s A + B for any B, By Prop, 1.25, we must show that any non-zero principal ideal of A + B has an R-semi-simple epimorph, and i t i s clear that we can restrict our attention to non-zero elements of the form a A b where a e A and b e B. But (A + B) . c A + B, and A , aftb a b a* which i s R-semi-simple, i s a retract of this 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 results for any radical, provided we re 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 for 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 cardinal, then KA i s unequivocal. Proof; Let X •= S(A) and suppose M i s a clopen subset of X - S(KA). Then M •» H(Ma: a <K) where MQ is a clopen subset of X and Ma •= X for a l l but a f i n i t e number of a. But then the partial product II(^; Mft = i s a retract of M and i s homeomorphic to XK, Algebraically, any non-zero principal ideal of KA has a retract isomorphic to KA. Thus the lower radical generated by any such principal ideal i s the same as the lower radical generated by KA. In other words, KA i s unequivocal. 6.22 Corollary: If A i s not i n a radical class R , then KA i s R-semi-simple for any i n f i n i t e tc. Proof; Since A has a non-radical epimorph A, i t cannot be in R. Being unequivocal, i t i s R-semi-simple. 6.23 Corollary; For any .a, there i s a superatomic algebra S such that HQS i s F Q-semi-simple. Proof; Let S be the f i n i t e - c o f i n i t e algebra on a set of cardinality ^a+1' s i n c e every non-zero principal ideal of #QS has cardinality Ma+lt i t must be F a-semi-simple. 6.24 Corollary; Every algebra i s a retract of an unequivocal algebra. The last corollary is a generalization of the fact 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 radical 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. For any proper radical class R, l e t o(R) <F Ha where a i s the least ordinal B such that F e £ R. Note that o(R) i s always i n f i n i t e , p 6.26 Theorem: Let R be a proper radical class and l e t A be an algebra with more than two elements. Then KA is R-semi-simple for a l l K > o(R), Proof: Let ic = tf > o(R). By §6, d) of the Preliminaries, KA has F 0 as a retract. By definition of a(R), F a t R, so KA £ R. By Cor. 6.22, KA i s R-semi-simple. 6.27 Examples: We l i s t c(R) for known R: i) a{0) - HQt i i ) o(P o) - j j o i i i i ) o(P) «fc 0, iv) o(E a) - f l a . v) o(F a) « ^ a + 1 , v i ) o<C o)'-Jf a, v i i ) o(6 a) - ftg where tfp - fy*0, v i i i ) o(J) -fy,, ix) o(K) - Hv CHAPTER SEVEN THE LATTICE OF RADICALS We can part i a l l y order Boolean algebra radicals by the relation 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 find that the Boolean algebra radicals form a l a t t i c e with some interesting algeb-raic properties, §1, Lattice-Theoretic Preliminaries 7.1 ; Definition: An abstract algebra ^L; v , A , *, 0, i s called a pseudo-complemented distributive l a t t i c e (with 0 and 1) i f <(L; v , A , 0, ]) i s a distributive l a t t i c e (with 0 and 1) and * is a unary operation on L satisfying a A b «* 0 i f and only i f b $ a*. Thus a* i s the maximum of the elements disjoint from a, A more general concept i s the following: 7.2 Definition: An abstract algebra ^ L ; v , A , 0, l ) i s a Brouwerian  lat t i c e i f i t is a l a t t i c e (with 0 and 1) in which, for 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, We denote the element c by (b:a), 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 distributive. In a complete l a t t i c e , an obvious candidate for (b:a) is the supremum of a l l x such that a A x 4 b. If the l a t t i c e also satisfies the i n f i n i t e distributive law: a A sup {a^: i e 1} « sup l a A a^: i e I}, then this w i l l suffice to show that the supremum c in questions does indeed satisfy 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 distributive l a t t i c e . Then for any a, b e L: a) i ) a .$ a**, i i ) a $ b implies b* < a*, i i i ) a* -• a***, iv) a o a** i f and only i f a ° b* for 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) Let L* «• {a*: a e L} » {a £ L: a =» a**}, called the skeleton of L» 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) Let D = {a e L: a* = 0), called the set of dense elements of L. 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 larger elements. 7.4 Lemma; Let p be an atom in a pseudo-complemented distributive l a t t i c e L. Then: i ) for any a e L, p ^  a V a*, and Proof: If p i s not contained i n a, then since i t i s an atom, p A a = 0. Then p < a* and i ) follows. If p a** but p i s not contained i n a, then again we get p a* so that p 4 a A a** «* 0, contradicting the fact that atoms are non-zero. The other direction of i i ) i s obvious, 7,5 Proposition: Let L be a complete pseudo-complemented distribu-tive 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 { a A t : a e L } « For any .a . cL, „def ine..a° = a* A ,t» .Then: i) \L f c; VpA, °, 0, y i s a Boolean algebra; i n fact, 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* for any a e L, Thus t = (a v a*> A t =• (a A t) v (a* A t) = (a A t) v a° for any a e L. If a e L f c, then a «= a A t, so we get t » a V a° for a l l a e L^ ., Clear-l y , a A a° » 0, so L^ . i s a Boolean algebra. Since i t i s complete and atomic, i t i s a power-set algebra. Define f: L * — L f c by f (a) « a A t. Then f clearly preserves A, Also, f (a*) » a* A t » a°. By Lemma 7,4, i i ) p a i f and only i f p 4 a**. a A p ** a** A p for a l l a e L. We extend this to t by showing that a A t » sup {a A p: p i s an atom). It i s clear that a A t i s an upper bound for this set. Suppose c i s any other upper bound. Then for any atom p, p £(a v a*) A (p v a*) = (a A p) V a* < c V a*. But then t $ c v a* and a A t « a A ( c v a*) « a A c ^  c. Hence a A t i s , in fact, the least upper bound of the set. It easily follows that a A t •» a** A t for a l l a e L. 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). Furthermore, for any a A t E Lfc, f (a**) = a A t. 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 in Prop, 7.5. Then L is 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 for a l l atoms p, so that a A t <= 0, But then a $ t* = 0, Conversely, i f L is 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 for any a e L, t «* (a A t) v/ (a* A t ) , we see that i f a e D, then t ^  a. Hence t i s a lower bound for D, and i n case t e D, then D i s the principal f i l t e r generated by t. In this case, L s p l i t s at t into a principal f i l t e r above t and a power-set algebra below i t . §2, The Lattice of Radicals for 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 radical for associative rings. The class of such radicals forms a complete l a t t i c e under the natural ordering. The meet of any collection of radicals i s their intersec-tion and the join i s the lower radical generated by their union. The join i s also determined by i t s semi-simple class, which i s the intersec-tion of the semi-simple classes of the radicals in the collection. The class of hereditary radicals forms a complete sublattice of the l a t t i c e of radicals and i s shown to satisfy an i n f i n i t e distributive law which makes the remarks following Defn. 7.2 pertinent. We con-clude that the l a t t i c e of hereditary radicals i s Brouwerian, distribu-tive, and pseudo-complemented. Snider shows that this l a t t i c e i s atomic, the atoms being the lower radicals generated by a single simple ring. Hence, using Prop, 7.5 and Prop. 7.6, we can extend his results as follows: 7.7 Proposition: Let T be the lower radical of associative rings gen-erated by the class of simple rings. Then: i) T i s hereditary and T* =• 0, i i ) the class of hereditary radicals contained i n T form a power-set Boolean algebra under the natural order, and i i i ) this 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 intuitively more obvious candidate for i t than he gives. Unfortunately, i t i s not, i n general, hereditary. It w i l l , however, yield a nice character-ization of (S:R) in universal classes for which every radical i s hereditary. 7.8 Proposition; Let R and S be hereditary radicals, l e t M be the class of R-rings which are S-semi-simple, and l e t W be U(M). If W i s hereditary, then W = (S:R), Proof; Since R i s hereditary and semi-simple classes are hereditary, the class M i s hereditary and so determines an upper radical. F i r s t , suppose A i s a ring i n R A W. R i s closed under epimorphs, so by definition of Wg A can.have-no nonrrzero.:»epimorph-which. is^S-semi-simple. But then A e S, so K A N S , Now suppose V i s a radical such that R \ V $ S . Let A be in M. Then 0 » s(A) r(A) n v(A) = A rj v(A) «= v(A). So A i s l/-semi-simple. Since W i s the largest radical for which M-rings are semi-simple, N l . Thus W = (S:R). 7.9 Corollary; Let R be a hereditary radical and l e t W be the upper radical generated by R, If W i s hereditary, then W = R*. Proof; R* « (0:R) i s the upper radical determined by R-rings which are semi-siiaple with respect to the zero radical; that i s , the class R, §3, The Lattice of Radicals for Boolean Algebras Snider's results can be appled immediately to radicals of Boolean rings, and since Boolean ring radicals are hereditary, we can use the descriptions of Prop. 7.8 and Cor. 7.9. 7.10 Proposition; The class of Boolean ring radicals forms a com-plete, Brouwerian, pseudo-complemented distributive l a t t i c e with ex-treme elements. If R and S are Boolean ring radicals, then (S:R) i s the upper radical generated by R-rings which are S-semi-simple, and R* i s the upper radical generated by R. We rec a l l that any non-zero radical class must contain the two-element Boolean algebra. Thus R A R* •» 0 entails that either R or R* must be 0, so that pseudo-complementation i s t r i v i a l , ^However, this same fact means that we can discard 0 and the Boolean ring radical L(2) w i l l serve as a zero for the new l a t t i c e . In order to see that the lattice-theoretic properties are essentially unchanged, a l l we need do i s verify the following: 7.11 Lemma: If R and S are non-zero Boolean ring radicals, then (S:R) i s non-zero. Proof: Since R and S are non-zero, 2 i s in both of them, and then R A L(2) - 1(2) $ S, Then L(2) $ (5:R). Then Prop, 7,10 holds without change for non-zero Boolean ring radicals, except for the description of pseudo-comple-mentSp which now becomes R* « (L(2_):R); that i s , R* i s now the upper radical generated by atomless R-rings, 7.12 The Isomorphism Theorem: Let Lat(E) be the class of non-zero radical classes of Boolean rings and Lat(A) the class of Boolean algebra radicals. Let f be the map which sends any non-zero Boolean ring radical R into the class of Boolean algebras in R. Then f i s a one-to-one correspondence between Lat(8) and Lat(A) which preserves order in both directions, and so <Lat ( B);v,A, *, L(2), 8> * <Lat(A); v , A , *, 0, A ) . Proof: By definition of radical classes .„o.f .algebras,, f .is onto.. Since we can recover R from f ( R ) as the class of Boolean rings, a l l of whose principal ideals are in f ( R ) , f i s one-to-one. The rest i s obvious. For the sake of completeness, we give a description of R* and R** for Boolean algebra radicals. 7.13 Proposition: Let R be a radical class of Boolean algebras. Then R* is the class of a l l algebras with no atomless epimorphs in R, and R** i s the class of algebras A such that any atomless epimorph of A has an atomless epimorph i n R. 7.14 Corollary; If A i s atomless, then 1(A)* < U(A). Proof; If B has no atomless epimorph in 1(A), then i t certainly has no principal ideal 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 radical class and A » H(A^j i e I) i s a product of R-algebras such that P(I) e R, then A.e Rft*. Recalling that P(I) i s a retract of A, l e t M be the ideal of A such that A/M = P(I). If J i s any ideal of A such that A/J i s atomless, l e t K(J) be the ideal 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 for any such J. (We use Pierce's result of the Preliminaries 3, j ) and the definition of R**.) We also present some considerations related to the conjecture that J «= 0. Suppose i s a class of homogeneous algebras closed under epimorphs (hence containing 2) and that ^ consists of a l l other homo-geneous algebras. Then any radical R for which H^-algebras are radi-cal and ^-algebras are semi-simple must satisfy L(H^) < R ^ UCf^). Taking H-^  = (2), we see that the conjecture J = 0 i s a special case of the conjecture that L(H^) ° U ^ ) . One can easily 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 in them, then (J:R) » (J:S). Thus we get: 7.15 Proposition: If J «• 0 and R and 5 are radical classes with the same homogeneous algebras in them, then R* «= S*, §4<> Dual Atoms and Complements in Lat(A) Snider's proof [25] that the l a t t i c e of hereditary radicals for associative rings has no dual atoms can be considerably simplified for Boolean algebra radicals. 7.16 Proposition: If R i s a proper radical class (that i s , not every algebra i s radical), then R i s properly contained i n a proper radical class. Proof: Since R cannot contain a l l free algebras, l e t a tea. some ordinal such that F a £ R. Then F f l E F f lV R, so this i s a radical class proper-ly containing R, For any 8 > a, F^ i s not in R or in F a» Being unequi-vocal, i t i s semi-simple with respect to both radicals, and so i t i s F a V R-semi-simple. Hence F aV R i s proper. Snider [26] gives a characterization of complemented heredi-tary radicals which we can use to deduce that 0 and A are the only complemented elements of Lat(A). We choose to deduce this from the following stronger result: 7.17 Proposition: The supremum of any set of proper radical classes i s proper. Proof: Let {R^: i e 1} be a set of radical classes. For each i e I, choose a(i) such that t Since I i s a set, the set of <x(i) has a supremum a. Then F^ is R^ - semi-simple for a l l i e I, and so i s semi-simple vit h respect to the supremum of these radicals, 7.18 Corollary: 0 and A are the only complemented elements of Lat (A), S5« Locating Known Radicals in Lat(A) We have already obtained some lattice-theoretic relation-ships between our radicals, and they w i l l not be repeated here as they are summarized in the diagram which comprises §7. 7.19 Proposition: For a l l a, Pa* - ?„• F * " Proof: Let V be any one of the following classes: atomless P a-algebras, for any a, or atomless P-algebras. Then V satisfies the conditions of Prop, 4.8 and so the upper radical determined by Y Is VQ. But R*, for any radical R, i s the upper radical generated by atomless R-algebras. Corollary: VQ i s in the Boolean algebra of skeletal elements of Lat(A), Furthermore, P « P * and for each a, P Q < VQ* 7.20 Proposition: Let B be the least ordinal such that exp Kg > exp//a Then 8 i s the least ordinal such that P„ •$ P e. In this case, the containment i s proper. Proof; By cardinality, P a does not have P^ as an epimorph, but a l l smaller power-set algebras are epimorphs of P A» If the radicals are equal, then VQ = VQ A » P Q A => P Q* A P Q = 0, which i s a contra-diction. 7.21 Proposition; Let g be the least ordinal such that exp /{^ > exp Then for a l l a >, B, t>a* - 0. Proof; For any a, we have VA* i VQ*. If a > 6, then by Prop. 7.20, ? o 4 4 t? a so that Va* i P Q* - P0.' But then Va* « 1?0 A P Q* - 0. 7 . 2 2 Corollary; Let B be as above, and l e t y » exp Hg. Then for • « • * Yt C a * = ° * : Proof; This follows., from the fact ..,that .£ ra J?g« u 7.23 Proposition; Let ftg - fra °. Then C a ^ G s $ C p i Proof; The f i r s t containment was proved in Prop. 4.6. Note that F c has F as an epimorph. Then i f A has F as a subalgebra, i t must, p a p by i n j e c t i v i t y of T t have F a as an epimorph. This, proves the second containment. 7.24 Corollary; Let y be as in Cor, 7,22. Then for a l l a £ y, Ga* - 0. 7.25 Proposition: Let « exp Ua» Then ?a « Fg, Proof; By cardinality, P^ E Fg, 7.26 Proposition: F a < c a + 1 « Proof: Clearly, F a e C a + 1 . 7.27 Corollary: F « V . *- a a Proof: Let H g « exp Ha } tfa+1. Then F a < C Q + 1 « C p - P Q. 7.28 Proposition: C .$ E . r 1 • 1 • a a Proof: F Q i s C0-semi-simple and E Q is the largest radical for which this i s true. 7.29 Corollary: F 0 < - E 0 + 1 . 7.30 Proposition: F * * E . Proof: Apply Cor. 7.14. 7.31 Proposition: For a > 0, E a* * 0. Proof: By Cor. 7.29, fQ £ E± $ E o. Then E a* 3 F Q*. By Prop. 7.30, Fo* * Eo * Ea« T h e n Ea* Ea» a n d s o Eo* " °* 7.32 Proposition: F Q* « E Q. Proof: By Prop. 7.30, F Q* •$ EQQ Suppose A does not have F Q as an epimorph, and l e t B be an atomless epimorph of A. If B e F f then i t has a countable principal ideal which would be isomorphic to F Q, contradicting the assumption on A, hence A has no atomless epimorph in F c; that i s , EQ <$ ¥Q. 7.33 Corollary: P Q* « E Q. Proof: By Cor. 7.27, F •$ P . — ' o o 7.34 Corollary: E Q i s in the Boolean algebra of skeletal elements of Lat (A). So also i s the radical E Q A P Q, which i s d i s -tinct from E . P , and 0. o * o Proof: The meet of skeletal elements i s skeletal. Note that F Q i s in VQ but not in E c, and P Q i s in E Q but not i n P 0, The algebra Q of Example 3,13 i s an atomless algebra i n E A P . o o 7.35 Proposition: K contains ? Q and F 0. K* * 0, If Hy = exp H D» then K £ F , and for a l l a J. Y» F„* *• 0, Y Proof: Since P Q arid F Q are separable, the f i r s t assertion follows immediately. The second statement i s simply a re-statenent of Theorem 3.6. Since every separable algebra has cardinality at most H t we get that K < F . Then i f a > y, K ^ F $ F « a n d Frv* K* = A» 7»36 Corollary: If y i s as in Prop. 7#35* and a £ y + 1 » then Proof: Use Prop, 7,26 and Prop. 7,35. This sharpens Cor. 7.22. 7.37 Corollary: If •= exp ft , then C a* = 0 £<$n? a l l a £ 2. Note that i n this case, C. » P , so Cor. 7.36 i s a best possible result. §6. Atoms in the Lattice 7.38 Theorem: Let R be an atom in Lat(A), Then: i ) for any non-superatomic algebra A e R , R = 1(A), i i ) there is an atomless. separable algebra A in R , such that R =• L(A), i i i ) any atomless algebra A in R i s unequivocal, and iv) for any atomless algebra A in R , L(A)* « U(A) so that U(A)* $ 0. Conversely, i f A is an atomless unequivocal algebra such that U(A)* & 0, then L (A) i s an atom in the l a t t i c e of radicals. Proof: The f i r s t assertion i s obvious, and i i ) follows from the fact that any..~no.n-8uper,a.tgmlc.,..algebra.*has„an.s,t,Pml.es.s6!.«separable«epimo,rph. 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, either 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 iv) follows. For the converse, suppose A is unequivocal and U(A)* ^ 0. 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 let -S be any radical and suppose A t S. Then A i s S-semi-simple, BO S 4 U (A) » 1(A)* and L (A) A S =» 0, 7.39 Theorem: P and F_ are atoms in Lat (A). 1 o o Proof: Write P = L (F N ) and F » L(F ). Then F and F are unequi-1 o N o ' o o o o M vocal. Also U ( F Q ) * * Vq* i 0, and U(F Q)* «= E D* 0. Hence, by the last theorem, ? Q and F Q are atoms. §7« A Diagram of the Lattice On the next page, we present a diagram of Lat(A), which summarizes the results of the last tVo sections. For simplicity, 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 radicals i n associative and alternative rings," Canad. J . Math. 17 (1965), 594-603. 2. Amitsur, S. A., "A general theory of radicals. II: Radicals i n rings and bicategories," Amer. J . Mathsi 76 (1954), 100-125. 3. Armendariz, E. P., "Closure properties in radical theory," Pacific J . Math. 2£;5(1968), 1-7. 4. Birkhoff, G,, Lattice Theory, 3rd edition, 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," Pacific J . Math. 15 (1965), 1145-1151. 7. — •,*" Super atomic Boolean algebras," Pacific J. Math. 23 (1967), 479-489. 8. Divinsky, N,, Rings and Radicals, University of Toronto Press, Toronto, 1965. 9. Easton, W, B., "Powers of regular cardinals," 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 di s -crete spaces i n bicompacta," Dokl. Akad. Nauk SSSR 189 (1969) = Soviet Math. Dokl. 10 (1969), 1391-1394. 12. Gillman, L., and Jerison, 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 Mathema-t 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 Tarski, A., "Measures in 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 distributive l a t t i c e s , I: Subdirect decomposition," Trans, AmerAcSoc. 156 (1971) , 335-342. 19. Leavitt, W. G., "Sets of radical classes," Publ, Math. Debrecen. 14 (1967), 321-324. 20. Monk, J, D., and Solovay, R,, M,, "On the number of complete Boolean algebras," Algebra Universalis 2 (1972), 365-368. 21. Mostovski, A., and Tarski, 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. «—- — f • V Some ^ questions *about"comlple te Boolean 'algebra s," Proc. Symp. Pure Math. 2 (1961), 129-140. 24. Sikorski, Roman, Boolean Algebras, 3rd edition, Springer-Verlag New York Inc., 1969. 25. Snider, R. L,, "Lattices of radicals," Pacific J . Math. 40 (1972) , 207-220. 26. —-— , "Complemented hereditary radicals," Bull. Austral. Math. Soc. 4 (1971), 307-320. 27. Yaqub, F. M., "Free extensions of Boolean algebras," Pacific J, Math. 13 (1963), 761-771. \ INDEX OF NOTATION Page references are given where they might be helpful. Ordinals: a, 8, Y t " . . . Well-ordered chains: {l a} (29), (h a(A)} (29), {Ra> (63). Cardinals: K, , |A|; K+ = next largest cardinal after K; exp K •= 2*.. Rings and Algebras: A, B, C, ... ; A = principal ideal of A generated by x; h*(A) (29); A « normal completion of the algebra A (10-11); 2_ =• two-element Boolean algebra; F q = free algebra on K 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 discrete topology; 2K"'** product of K copies of H the two-element discrete space, a Cantor space; 2 ° the Cantor set; 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-ras: E(A : i e I), A + B (12), KA (75); product of toplogi-cal spaces: II(X±: i e I), X <. Lattice 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, ... ; radical classes with corresponding radical ideal: R, r(A), S, s(A), ... (18-19); lower radical: L(X) (28); upper radical U (/) (45); 8 - the class of Boolean rings; A • the class of Boolean algebras; Lat(B) « the class of radical classes of Boolean rings; Lat(A) « the class of radical classes of Boolean algebras. Radical Classes of Boolean Algebras: 0 ** the superatomic Boolean algebras (36), F a = lower radical generated by F & (38), K » lower radical generated by separable algebras (40), P Q • lower radical generated by P a (40), P = lower radical generated by complete algebras (42), E Q => upper radical determined by F a (48), G a = upper radical determined by F Q (48), J <= upper radical determined by atomless homogeneous algebras (50), J v «= upper radical determined by atomless v-homogen-enous algebras (51), C Q » algebras without F a as a subalgebra (53), VA " algebras without P as an epimorph (54). 

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