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C. C. Chen and G. Grätzer, on the Construction of Complemented Reprinted from JOURNAL OF AWE8RA Vol. II, No, 1, January 1969 All Rights Reserved by Academic Pres., New York and London Printed in Belgium On the Construction of Complemented lattices C. C. CHEN AND G. GRATZER* Department of Mathematics, The University of Manitoba, Winnipeg, Canada Department of Mathematics, Nanyang Umversity, Singapore Communicated by Saunders MacLane Received Dec. 3, 1967 1. INTRODUCTION R. P. Dilworth's classical resultl that every lattice can be embedded in a uniquely complemented lattice is well known but the proof itself is in­ accessible to the general reader. The reason for this is a simple one: the proof is too specialized, in the sense that a difficult machinery has to be developed for which no other application has been found. We suggest that all the deeper results developed by Dilworth can be avoided. As the first step of the proof let us observe that it suffices to show that an at most uniquely complemented lattice L o can be embedded in a lattice lv[ of the same type in such a way that 0 and 1 are preserved and such that each element of Lo has a complement in M. The second step consists in applying a method of Dilworth (Theorem 2.2. of UC) for the description of the free lattice L generated by a partially ordered set obtained from Lo by adjoining an unordered set S. Finally, if S is in one-to-one correspondence with the non-complemented elements ofL o , then the method of singular elements used in the final step of Dilworth's proof can be applied directly to yield a subset M ofL, which is a lattice and which has the properties required in the first step. The simplicity of this approach makes it possible to prove some far reaching generalizations. It should be emphasized that we not only get the embedding theorem but we also get the solution to the word problem in the free uniquely comple­ mented lattice, as in UC. * This research was supported by National Research Council of Canada. 1 R. P. Dilworth, Lattices with unique complements, Trans. Amer. Math. Soc. 57 (1945), 123-154; this paper will be referred to as UC. 56 57 ON THE CONSTRUCTION OF COMPLEMl>NTED LATIICES 2. RESULTS A lattice L is called at most uniquely complemented if every element of L has at most one complement; L is uniquely complemented if every element has exactly one complement. THEOREM 1. Every at most uniquely complemented lattice can be embedded in a uniquely complemented lattice with 0 and 1 preserved. Since every lattice can be embedded into an at most uniquely complemented lattice, Theorem 1 yields Dilworth's result. Let m be a cardinal; a latticeL is anm-eomplemented(resp., m-complemented) lattice, if every element of L, different from 0 and 1, has exactly (resp., at most) m complements. THEOREM 2. Let L be an m-complemented lattice with incomparable complements (i.e. if b, c are complements of a, then b ;::?: c implies b c). Then L can be embedded in an m-complemented lattice with 0 and 1 preserved. DEFINITION 1. The lattice L is the free tn-complemented lattice on n generators if it satisfies the following properties: (i) L is an m-complemented lattice; (ii) L contains a subset G of cardinality n such that the smallest sublattice of L which is closed under complementation and which contains G isL; (iii) for any m--eomplemented lattice K, and for any map rp of G (as given in (ii» into K, there exists a lattice homomorphism ip of L into K such that ip extends rp and Lip (the image ofLunder ip), as a sublattice of K, is closed under complementation in K. THEOREM 3. For any (non-zero) cardinalsm andn, thefree m-complemented lattice on n generators exists and it is unique up to isomorphism. A special kind of (complemented) lattices, typified by the five element modular, non-distributive lattice, are the homogeneous lattices: The lattice L is called homogeneous if the relation a"" b; "a = b or a is a complement of b" is an equivalence relation. Note that every lattice can be embedded in a homogeneous lattice (by adding a new 0 and a new 1) and that any sublattice of a homogeneous lattice, containing 0 and 1, is again homogeneous. THEOREM 4. Every m-complemented homogeneous lattice can be embedded in an m-complemented homogeneous lattice. CHEN AND GRATZER 58 By adding the adjective "homogeneous" wherever necessary in the definition of "free m-complemented lattice on tt generators," we get the definition of ''free homogeneous m-complemented lattice on tt generators." THEOREM 5. For any (non.zero) cardinals m, tt, the free m-complemented homogeneous lattice on n generators exists and it is unique up to isomorphism. It may be of some interest to point out that the extensions of a map of the "generators" to a homomorphism is not unique (see Section 6). Thus the uniqueness statements in Theorems 3 and 5 are not as obvious as the unique. ness statement is for free algebras in the ordinary sense. 3. A VARIANT ON A THEME BY DILWORTH Let Lo be a lattice with 0 and 1; lattice operations will be denoted by v and A. Let S be a set disjoint to Lo ; we make Q = Lo V S a partially ordered set by defining x ::;;;: y in Q if x = y or x, y E Lo and x::;;;: y in Lo. DEFINITION 2. Let S(Q, V, fi) be the set of all finite sequences whose entries are elements ofQ, and v, ii, (,). The set P(Q) of (lattice) polynomials over Q is the smallest subset of S(Q, V, fi) such that (i) Q k:. P(Q); (ii) if A, B EP(Q), then (A V B), (A fi B) E P(Q). The parentheses will be omitted whenever there is no danger of confusion. DEFINITION 3. For some A E P(Q) we define the upper cover A of A as follows: (i) for A ELo , A = A; (il) if A = Bo V B1 , then A exists iff 11., and 111 exist and A = 110 V 111 ; (iii) if A = Bofi B1 , then A exists iff 110 or B1 exists; A = 110 A 111 if both exist, and A = 11i if 11$ (i =f: j) does not exist. Dually, we define 4 the lower cover of A. Remark. Intuitively, A is the smallest element of Lo containing A in the lattice freely generated by Q. Note that A and 4 are always in Lo, and if both exist, then A :?= 4. DEFINITION 4. Set A C B; for A, B E P(Q) if it follows from the rules (1)-(6) below: (1) .A = B; (2) A and lJ existand A ~ lJ (in~); (3) A = All V At , Aok:. B and At C B 59 ON THE CONSTRUCTION OF COMPLEMENTED LATTICES (4) A = Ao (\ A1 ,Ao k B or A1 k B; (5) B = Bo U B1 , A k Bo or A k B1 ; (6) B = Bo (\ B1 , A k Bo and A k B1 • Set A - B, if A k Band B k A. THEOREM A. The relation is an equivalence relation. For A E P(Q), let (A) denote the equivalence class containing A, andputL = {(A) I A eP(Q)}. Define (A) ~ (B) iff A k B. This makes L into a lattice. Identifying A EQ with (A), we have Q CL; in fact, L is the lattice freely generated by Q. Finally, tJ (resp., A) is the largest (resp., S1nallest) element of Lo contained in (resp., containing) A, and tJ (resp., A) e.msts ifand only if0 ~ A (resp. A ~ I). This theorem is contained in Theorem 2.2 of UC if the cardinality of S does not exceed the cardinality ofLo . Since the proof of Theorem 2.2 of UC applies without any essential change we refer the reader to UC. It follows from Theorem A that (1)-(6) apply to ~ in L. Since there is no danger of confusion we zvill write A for (A). 4. THE BAste. LEMMA Let Lo, S, Q and L be given as in Sectt'on 3. Let C(Lo) denote the set of all two element sets {x, y} such that x is a complement ofy in Lo• Let C denote an arbitrary set of two element sets {x,y} such that x 0:/= y, x, y EQ and ifx ELo ' then x 0,1 and y ¢:Lo (i.e. either x,y E S, or x ELo , x 0,1 and YES, or the symmetric case). DEFINITION 5. Callan element A ELjoin-singular, if 1 ~ A or x,y ~ A for some {x, y} E C; meet-singular, if A 0 or A ~ x, y for some {x, y} E C. The element A is singular if it is either join-, or meet-singular. Compare this definition with Definition 4.1 of UC. DEFINITION 6. The only component of A E Q is A. The components of Au B (resp. A (\ B) are Au B (resp. A (\ B) and the components of A and B. Now we are ready to define a new lattice M: M is the set of all those A, A E L, for which A has no singular component, and two new elements u and z. For these new elements the order is given by u ~ X ~ z for all X EM. For the other elements X ~ Y in M if and only if X Y inL.
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