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DIRECT DECOMPOSITIONS WITH FINITE,DIMENSIONAL FACTORS. PETER CRAWLEY, The principal results A fundamental theorem of Ore 10 states. that if an element in a finite dimensional modular lattice is represented. in two ways as a direct join of indecomposable elements then the factors. of the two decompositions are projective in pairs The Krull Schmidt. theorem is an immediate consequence of this result Subsequently many. authors have considered direct decompositions in modular lattices In. particular Kurosh 8 9 and Baer 1 2 obtained conditions which imply. the existence of projective refinements of two direct decompositions of. an element in an upper continuous modular lattice When applied to. the decompositions of a group G the conditions of Kurosh and Baer. are reflected in certain chain conditions on the center of G In a somewhat. different direction Zassenhaus 11 has shown that the representation. of an operator group as a direct product of arbitrarily many indecomposable. groups each with a principal series is unique up to isomorphism. This paper studies the direct decompositions of an element in an. upper continuous modular lattice under the assumption that the element. has at least one decomposition with finite dimensional factors It is. then shown that every other decomposition of the element refines to. one with finite dimensional factors and that a strong exchange isomorphism. exists between two decompositions with indecomposable factors This. latter result sharpens the uniqueness result of Zassenhaus. Before explicitly stating the principal results let us note the following. definitions A lattice L is upper continuous if L is complete and. for every element aeL and every chain of elements xk ke K in L. If a and a i e I are elements of a complete lattice L with a null. element 0 then a is said to be a direct join of the elements a i e I. in symbols, if a Jiei aif and for each index he I we have ah J h ai 0 The. direct join of finitely many elements al9 is also denoted by. ax U U An element b is called a direct factor of a if a b U x. for some element x An element a is indecomposable if a 0 and a. Received July 7 1961 This work was supported by the Office of Naval Research. 458 PETER CRAWLEY, x U y implies x 0 or y 0 Finally an element a is said to be finite. dimensional if every chain of elements less than a is finite. THEOREM 1 If a is an element of an upper continuous modular. lattice and,a x J y Jti, where x is finite dimensional and indecomposable then there exists an.
index he I such that th r U s and, THEOREM 2 If a is an element of an upper continuous modular. lattice and a is a direct join of finite dimensional elements then every. direct factor of a is also a direct join of finite dimensional elements. THEOREM 3 If a is an element of an upper continuous modular. lattice and,a U ai U i, where each a i e I and each b j e J is finite dimensional and. indecomposable then there is a one to one mapping of I onto J such. a a U 0 bj,for each index iei, These theorems may be applied directly to the lattice of admissible. normal subgroups of an operator group to yield the following extension. of the result of Zassenhaus mentioned above If an operator group G. is a direct product G ILe 4 where each of the factors A i e I has. a principal series then any two direct decompositions of G have. centrally isomorphic refinements, Even with the strong continuity assumption it seems impossible to. relax the assumption of finite dimensionality particularly in Theorems 1. and 3 The free abelian group of rank 2 shows that in general Theorems. 1 and 3 fail for lattices satisfying only the ascending chain condition. The example in the following paragraph shows that continuity and the. descending chain condition also are not sufficient for these results It. Cf J nsson and Tarski 6, DIRECT DECOMPOSITIONS WITH FINITE DIMENSIONAL FACTORS 459.
is curious that Theorems 1 and 3 hold for groups whose normal subgroup. lattices satisfy only the descending chain condition1 and yet fail for. general continuous modular lattices satisfying the descending chain. The example is as follows Let p be an odd prime and let G be. an additive abelian group isomorphic with,Z p x Z p x Z p x Z p x Z p x Z p. where Z p denotes the cyclic group of order p and Zip00 denotes the. generalized cyclic p group Let Q R S T U and V be subgroups of. G with Q R Z p S T U V Z p and,G Q j S U T Z7LJ V, Let q and r generate Q and R respectively and let S T Uy and V be. generated respectively by sets sn n un 9 and vn where. psx 0 psn 1 sn n 1 2, with analogous relations holding for the tn s un s and vn Set A. Q U S U T and B R U U U F Let C be the subgroup generated by. the set q r s2 s2 s3 u29 t v2 tlf v3 2 and let D, be the subgroup generated by q 2r i 2 su uz s2 x a. i 2 It then follows that,and G A j B A U C A l D B C B U D C D Further.
more A C An D B iC B n D O Now let L be the set of all. subgroups I g S U U U U F together with all subgroups of the form. i U l S U l C U l a n d f l U l with X g S U T U C U V and the. group G It is easily checked that under set inclusion the elements of. L form a complete sublattice of the lattice of all subgroups of G Hence. L is an upper continuous modular lattice satisfying the descending chain. condition Moreover the subgroups A By C and D are indecomposable. in L and each is protective only with itself Thus Theorems 1 and 3. fail for the direct joins G i U C U, Proofs of the theorems The usual notation and terminology is. used throughout Lattice join meet inclusion and proper inclusion are. denoted respectively by U S and If and 6 are elements of. a lattice and b g then the quotient sublattice x b x is denoted. by The symbol s denotes the isomorphism of two lattices The. null element of a lattice is always denoted by 0, We begin with the following lemmas The first is generally known. 460 PETER CRAWLEY, LEMMA 1 If L is an upper continuous lattice S is a subset of. L and a is any element of L then,anus u a n U, where j is the collection of all finite subsets of S. The lemma is trivial when S is finite Suppose that S is an infinite. subset of L and suppose that the lemma is true for every subset S. of cardinality less than the cardinality of S Then there is a chain Si. ie I of subsets of S such that each Si has cardinality less than that. of S and such that S is the set sum of the subsets Si iel If t. is the collection of all finite subsets of Si9 applying upper continuity. and the inductive assumption we therefore have,a n U S a U U SJ U a U St.
U U n U U nUf,iei Fe i Fe,and hence the lemma follows by induction. An element c in a complete lattice L is said to be compact if for. every subset S C L with c g J S there is a finite subset S S S such. that c g U S A lattice L is compactly generated if L is complete and. every element of L is a join of compact elements 2 The next lemma is. an immediate consequnce of the definition of compactness. LEMMA 2 If cly cn is a finite set of compact elements in a. complete lattice then cx U U cn is also compact, 3 Every finite dimensional element in an upper continuous. lattice is compact, We shall first show that if q is completely join irreducible then q. is compact Suppose SQL and q J S Let p U x x q Then. p q since q is completely join irreducible Let j denote the collection. of all finite subsets of S If J F Q for every Fe then. g U d hence q f J F p for every F e J And it follows. by Lemma 1 that,q q U S U Q U P,a contradiction Hence q is compact. Now suppose that a is a finite dimensional element different from. For a discussion of compactly generated lattices see 4. DIRECT DECOMPOSITIONS WITH FINITE DIMENSIONAL FACTORS 461. 0 and suppose that every element properly contained in a is compact. If a is join irreducible then a is compact from above If a is not join. irreducible then there are two elements b c a such that a b U c. Since b and c are compact a is therefore compact and the lemma follows. by induction, LEMMA 4 If an element a of an upper continuous modular lattice.
is a join of finite dimensional elements then the quotient sublattice 0. is compactly generated and each compact element is finite dimensional. For suppose a J C where each c e C is finite dimensional If. x 5 then with j denoting the set of all finite subsets of C we have. n u c u x n U, Since the lattice is modular x U F is finite dimensional and hence. compact for each Fej The lemma now follows, LEMMA 5 If c au 2 an are elements of a compactly generated. lattice c is compact and c ax U U then for each m 1 n. there is a compact element dm S am such that c d J J dn. Since the lattice is compactly generated for each m 1 n there. is a set Cm of compact elements such that am U Cm Then c. U C U U U Cn9 and since c is compact there are finite subsets. C m S Cm such that c U C U U J C n By Lemma 2 U C is a. compact element for each m 1 n, LEMMA 6 a x y are elements of a modular lattice x J y. U w j g OJ U y then a x J a y,For a a 2 x 1 0 and a U y a U y. Proof of Theorem 1 Suppose x y ti i e I are elements. of an upper continuous modular lattice x is finite dimensional and. indecomposable and, Since x is compact by Lemma 2 there is a finite subset of indices.
ii i j S such that a th t n For each m 1 2 n,let us set. 4 l U U V l U i B 1 U tin,462 PETER CRAWLEY,and define xm x U tm tim Then it follows that 3. Now x m n U t J B1 n tm t f 0 and,U m a U O n tim U t t U L a U J a U. Thus m 0 m tfm n l U fm TO x U W Fm x x n TO and hence. each xm is finite dimensional and its dimension does not exceed the. dimension of x It follows that b x U U xn is finite dimensional. Since x 6 x U 2 we infer from Lemma 6 that,6 x U 6 n y. Therefore since is indecomposable and the dimension of each xm is at. most the dimension of x it follows from Ore s theorem that renumbering. the xm s if necessary,b x1 b y x i x3i xn, Then 2 U x y U y U 2 U b a From the fact that xx is.
finite dimensional and xjx1 y U 2 l y I b y b y b xJO it. follows that c i 2 0 Thus,Moreover since xx it follows from Lemma 6 that. th u y n t,Let us set,T h e n since x 2U U w a U U t 1 w e have. u n 4l j a u x u u n u y n u t x,6 U 2 4 l U x 4 l U i f a. and since a a y 4l U tfj a y n 4l U ifx i 0 a 0 it follows. that 2 U J 0 Hence,a L n 4 l U t J x U 2 U U ti,See for example 3 p 95. Actually we use the somewhat stronger version of Ore s theorem given in 5 pp 128. DIRECT DECOMPOSITIONS WITH FINITE DIMENSIONAL FACTORS 463. and the proof of Theorem 1 is complete, Proof of Theorem 2 Throughout the proof of Theorem 2 we will.
assume that a is an element of an upper continuous modular lattice and. where each ai i e I is finite dimensional and indecomposable. Suppose a r U s We shall first show that r and s are direct. joins of elements which are joins of a countable number of compact. Consider the collection of all subsets P of the lattice which. satisfy the following conditions,for some subset K I. 2 t r 0 t s for each t e P, 3 r and l s are both joins of a countable number of compact. elements for each te P, is nonempty since the null set is in m Moreover since by Lemma. 1 a set is independent if every finite subset is independent it follows. that the set sum of a chain of sets in also belongs to By the. Maximal Principle contains a maximal element Q,g L J Q L U w t n r v n. eM teQ teQ, Then it follows from condition 2 that q u U v and from condition.
1 that a q Jb u Jv Jb where b U e Furthermore if, we set r r 6 U v and s s U v then it follows from Lemma. 6 that r r J u and s s U v Hence, Suppose q a Then for some ioel we must have aiQS Q Since. is compact and 0 is compactly generated by Lemma 3 it follows by. Lemma 5 that compact elements c rr and d s exist such that. aio cx U dx U g, U tZx is also compact and hence there is a finite subset M S I such. The proof of this part was suggested by the main theorem of Kaplansky 7. 464 PETER CRAWLEY, Again JieMlai is compact and hence there are compact elements c2 rr. and d2 s such that,U 2 U d2 U g, Continuing in this way we get a sequence i0 Mu M2 fn.
of finite subsets of I and two sequences of compact elements. Ci c2 cn r and cix d2 dn g s such that,U a i n 1 U d i U. for each n 1 2, Then r g r and s s and if M is the set sum of the sets M i0. Mu M2 it is clear that,t u t U g U, Hence the set sum of and is a member of properly containing. Q Since this is contrary to the maximality of Q we must have q. It follows that r u and s v and thus r and s are direct joins of. elements which are joins of a countable number of compact elements. We now prove the following if is a direct factor of a and c is. a compact element with c 6 then there exists a finite dimensional. direct factor w of b such that c w Suppose 6 LJ e Since c is. compact there is a finite subset iu in s such that. U U tn c Applying Theorem 1 to the element ai and the. decompositions,a ai LJ LJ 6 LJ, it follows that b b J b e e J e where either b 0 or 0. a b LJ LJ LJ LJ LJ ej, DIRECT DECOMPOSITIONS WITH FINITE DIMENSIONAL FACTORS 465.
Now consider the direct decompositions, If we apply Theorem 1 to the element ah and these decompositions then. since ah U 2 0 it follows that j 63 U 65 ej 62 0 e. The principal results A fundamental theorem of Ore 10 states that if an element in a finite dimensional modular lattice is represented in two ways as a direct join of indecomposable elements then the factors of the two decompositions are projective in pairs The Krull Schmidt theorem is an immediate consequence of this result Subsequently many authors have considered direct decompositions