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Cardinality questions concerning semilattices of finite breadth
Discrete Mathematics 48 (1984) 47-59 North-Holland
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CARDINALITY QUESTIONS CONCERNING SEMILA.TrICES OF F I N r r E B R E A D T H S.Z. DITOR* Department of Mathematics, University of Western Ontar/o, London, Ontar/o, N 6 A 5B9, Canada
Received 22 September 1978 Revised 18 July 1983 It is shown that if L is a lattice in which every element has only finitely many predecessors and (*) every element has no more than k immediate predecessors, for some positive integer k, then I L l ~ k - r An example is constructed in which k = 2 and ILl=~t, but the question of whether ILl =~k-~ is possible for k > 2 is left unanswered. The conclusion ILl~
1. Introduction Let A be a set and ~:(A) the finite subsets of A ordered by inclusion. R. Haydon raised the following question concerning ~(A). D o e s t h e r e e x i s t a c o f i n a l s u b f a m i l y ~:0 (i.e., e v e r y G in ~ ( A ) is a s u b s e t o f s o m e F in ~ o ) w h i c h is c l o s e d u n d e r f i n i t e i n t e r s e c t i o n a n d is s u c h t h a t , f o r e v e r y F in ~ 0 , t h e c o l l e c t i o n o f p r o p e r s u b s e t s o f F w h i c h a r e in ~:o h a s at m o s t t w o maximal
members
(i.e., e v e r y
~:o-Set h a s
at m o s t
two maximal
proper
~0-
subsets)? Haydon
a s k e d this q u e s t i o n ( a n d a n s w e r e d it f o r t h e c a s e [A] ~
tion with a certain topology problem concerning the space of probability measures o n t h e p r o d u c t {0, 1} A, w h o s e r e p r e s e n t a t i o n as a n i n v e r s e l i m i t o v e r t h e i n d e x s e t ~:(A) offered an approach to solving the problem. When
A is c o u n t a b l e , t h e r e is n o difficulty in f i n d i n g s u c h a s u b f a m i l y ~ o o f
* This research was supported in part by a grant from the Natural Sciences and Engineering Research Council Canada. 0012-365X/84/$3,00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
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S.Z. Ditor
~ ( A ) . For, if A = { a 1, a 2. . . . }, we can take ~r0={F1, F 2. . . . }, where / 7 , = {a 1. . . . . a,}. It is when A is uncountable that the question becomes interesting. In this paper, we shall reformulate this question about ~r(A) in the setting of lattices. H e r e the essential elements of the p r o b l e m present themselves with greater clarity and several generalizations immediately b e c o m e apparent. A corollary of one of the theorems we p r o v e provides a s o m e w h a t surprising answer to the question raised by Haydon. W e show that if L is a lattice in which every element has only finitely m a n y predecessors and no m o r e than k immediate predecessors, for some positive integer k, then L has cardinality ILl~<~k_l. In addition, we provide an example in which k = 2 and ILl = R1. It follows f r o m these results that H a y d o n ' s subfamily ~:o exists if and only if IAI ~ 2 , or do we always have W e present two proofs of the main result on u p p e r bounds for the cardinality of semilattices (in fact, of directed sets) that have finite breadth. O n e is direct and the other is hn application of a t h e o r e m of ErdSs and Hajnal [2] on set-mappings, which, in turn, is based on a t h e o r e m of Kuratowski and Sierpinski [3]. T h e author is indebted to P. Erd6s for bringing [2] to his attention. In adtlition, the author would like to thank R.J. Koch for several useful conversations during the course of this research.
2. Preliminaries Let us first recall a few definitions and introduce the terminology we shall be using. A poset (partially ordered set) is a set P equipped with a reflexive, antisymmetric, and transitive relation ~<. If, for all x, y in P, either x ~< y or y ~
y<~x for some x ~ X } ,
which is written P[x] when X = then X is said to be colinal in If y < x and for no z in P is the set of elements covered by
{x}. T h e upper set of X is P*[X]. W h e n P [ X ] = P, P. y < z < x, then we say that x covers y. W e denote x by cov(x):
coy(x) = {y: x covers y}. A poset P is said to be directed upward (resp. directed downward) if every pair of elements x, y in P has an upper bound (resp. lower bound); if there is a least upper bound, or sup, x v y (resp. greatest lower bound, or irff, x A y ) , then P is
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called a join-semilattice (resp. meet-semilattice). When both x v y and x ^ y exist for all x, y in P, then P is called a lattice. If P is a join-semilattice, then the operation x v y is associative, commutative, and idempotent and for any finite sequence x 1. . . . . x~ in P, the least upper bound sup{x1 . . . . . x,} exists and coincides with x l v - - - v x ~ , which we shall sometimes denote simply by V x~ (when the domain of i is understood). A similar statement holds for meet-semilattices. A subset M of a lattice P is called a sublattice if x v y and x ^ y are in M whenever x and y are in M, and a lattice subset if the poset M (with the ordering inherited from P) is a lattice. If P is a join-semilattice, then a subset I is called an ideal if x v y ~ I whenever x, y ~ L and P [ / ] = L W h e n P is a meet-semilattice, the condition P[I] = I is equivalent to r ^ x ~ 1 whenever r ~ P and x ~/. If a poset P has a smallest d e m e n t , we normally denote it by 0 and say that P has a 0. Of course, if P has a 0, then P is directed downward. Suppose, on the other hand, that the non-empty poset P is directed downward but has no 0. Then, given any xl ~ P, there exists y ~ P such that x~ ~ P [ y ] . If, now, x2 is a lower bound for {x~, y}, then we must have x 2 < xx. Continuing, we can next find x 3 < x 2, and so on. Hence, we see that a poset which is directed downward either has a 0 or contains a strictly decreasing infinite sequence.
3. Lower finite lattices and k-lattices W e shall call a poset P lower finite if P[x] is finite for all x in P. If P is lower finite and the function [cov(x)l is bounded on P with maximum value k, then we shall say that P is a k-poset. A k-poset which is also a lattice shall be called a k-lattice. If to is the set of natural numbers with their usual order, then, clearly, to is a I-lattice and to xto x - - . xto (k factors) with the product order ( ( n i ) ~ ( m l ) if and only if ni ~~2. It follows from this that L is a l-lattice if and only if L is a lower finite chain, and this clearly is equivalent to L being order isomorphic to either a finite ordinal or to. Therefore, a I-lattice is always countable. More generally, as we shall verify in Section 5, if L is a k-lattice, then ILl ~
S.Z. Direr
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3.2. l~.Y~mple, d~(A), the m o n o i d g e n e r a t e d by the set A , consists of all finite sequences in A , including the e m p t y s e q u e n c e 0. T h e p r o d u c t of the two sequences s and t is the s e q u e n c e st o b t a i n e d by c o n c a t e n a t i o n , with ~ acting as an identity. If we define s ~< t to m e a n that t = usv for s o m e u a n d v, then J~t(A) b e c o m e s a lower finite poset which is directed u p w a r d a n d d o w n w a r d (¢ ~< s for all s). If a~ . . . . . a , e A and s = a 1 • • • a,, then cov(s) = {a 1 • • • a , _ 1, a 2 • • • a,}, so f o r IA[~>2, ~ ( A ) is a 2-poset. H o w e v e r , for IA[~>2, ~ t ( A ) is n o t a lattice since f o r distinct elements a, b in A , {ab, ba} has only ¢, a, and b as lower b o u n d s a n d therefore has no greatest lower b o u n d . W h e n A is infinite, [~t(A)l = IA[. 3.3. E x a m p l e . L e t S be the subfamily of ~ : ( A ) consisting of 0 a n d all the singletons {a}, a c A . T h e n S is a 1-poset which is a meet-semilattice a n d
ISI--IAl+l. T h e situations exemplified a b o v e c a n n o t be significantly e x p a n d e d b e c a u s e of the condition of lower finiteness. F o r example, if P is a lower finite joinsemilattice which is directed d o w n w a r d , then P is a lattice, since for any x, y in P, the set I of lower b o u n d s for {x, y} is a finite, n o n e m p t y ideal a n d t h e r e f o r e its least u p p e r b o u n d sup I exists and belongs t o / , i.e., x A y = sup L M o r e o v e r , by o u r earlier remarks, a n o n - e m p t y lower finite p o s e t is directed d o w n w a r d i~ and only if it has a 0. This proves the first part of the following proposition. 3.4. Proposition. (1) I f P is a lower finite ]oin-semilattice, then, with the ad]unction
of a 0 if necessary, P is a lattice. (2) I f P is a lower finite meet-semilattice which is directed upward, then P is a lattice. P r o o f . Suppose P x, y ~ P. W e must m e m b e r . Since S n o n - e m p t y subset l e m m a [4, p. 44].
is a lower finite meet-semilattice which is directed u p w a r d a n d show that S, the set of u p p e r b o u n d s for {x, y}, has a smallest is n o n - e m p t y a n d contains inf F = x~ ^ . • . ^ x , f o r every finite F = {x~ . . . . . x n} of S, t h e p r o o f of (2) follows f r o m the following
3.5. I~mmR. [_f p is a meet-semilattice in which every strictly decreasing sequence is finite and X is any non-empty subset of P, then inf X, the greatest lower bound of X, exists; moreover, inf X = xl A" • • A X, for some finite subset {x~ . . . . . x,} of X. l h ~ o L S u p p o s e the l e m m a is false. T h e n , for x~ . . . . . x r e X , x ~ ^ - • • ^ x r is not the greatest lower b o u n d of X, and so, t h e r e exists x , + ~ X such that x~ A" • • A Xr A Xr+~ < X1 A" • • ^ X,. It follows that S contains a strictly decreasing infinite sequence, c o n t r a r y to o u r hypothesis. T h e next proposition is n o w immediate.
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3.6. P r o p o s i t i o n . If ~;o is a cofinal subfamily of ~;( A ) and ~o is closed under finite
intersection, then ~:o is closed under arbitrary intersection and is a lower finite lattice with respect to inclusion. Conversely, if L is a lower finite lattice, then x --~ L[x] is an order-isomorphism (in fact, a meet-isomorphism) of L onto a cofinal subfamily of ~:(L) which is closed under intersection. L e t us return n o w to H a y d o n ' s question concerning ~r(A). If A is infinite and ~ o is cofinal in ~ ( A ) , then A = U ~0, f r o m which it is easily verified that [AI = I~r01. It t h e r e f o r e follows f r o m 3.6 and o u r previous r e m a r k s that if K ~>R 1, then H a y d o n ' s subfamily ~r0 exists for [A[ = K if and only if t h e r e exists a 2-lattice L with [ L I = K. T h e t h e o r e m that we shall p r o v e regarding k-lattices actually pertains to a larger class of semilattices, which we shall n o w consider.
4. Breadth In a join-semilattice S, a finite s e q u e n c e {x~}~=1 of length n is called redundant if t h e r e exists an index ] such that V x ~ = V ~ . j x ~ (equivalently, x ~ < V ~ . j x ~ ) ; otherwise, it is called irredundant. The sup of the lengths of the i r r e d u n d a n t sequences in S is called the breadth of S and is d e n o t e d by b(S). F o r e x a m p l e , if b ( S ) = 1, then S is a chain, since for all x, y in S, x v y is either x or y. 4.1. P r o p o s i t i o n . I l L is a k-lanice, then b(L)<~k. l~root. L e t L be a k-lattice and {x~}~__+~an arbitrary s e q u e n c e of length k + 1 in L. It suffices to show that V x~ = V~.ix~ for s o m e index j. L e t y = V x~, Yi = V~.jx~ (j = 1 . . . . . k + 1), and s u p p o s e that yj ~ y for all ]. T h e n Yi < Y for ] = 1 . . . . . k + 1 a n d for each j t h e r e exists z~ ~ cov(y) such that yj ~< z r Since Icov(y)l ~< k t h e r e exist distinct indices l, m such that zt = z m. H e n c e , yl v ym ~ 2 and L consists of the sets ¢, {1} . . . . , {k}, {1 . . . . . k}, o r d e r e d by inclusion, then L is a k-lattice and b ( L ) = 2. (2) T h e r e exist l o w e r finite lattices of finite b r e a d t h which are not k-lattices. F o r e x a m p l e , if L consists of the e m p t y set, the singletons {n}, and the sets {1 . . . . . n2}, for n = 1, 2, 3 . . . . . o r d e r e d by inclusion, then it is easily verified that L is a lower finite lattice of b r e a d t h 2 but is not a k-lattice. In purely algebraic terms, a semilattice is a set S on which t h e r e is defined a binary o p e r a t i o n (product) which is associative, c o m m u t a t i v e , and i d e m p o t e n t .
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T h e join-semilattice d e t e r m i n e d by S is o b t a i n e d by interpreting the p r o d u c t xy as the join operation, which a m o u n t s to defining x ~~n. T h e r e f o r e , b ( S ) ~ b s ( P ) . W e n o w e x t e n d the notion of b r e a d t h to any poset P by defining a finite s e q u e n c e {xl}~= 1 of length n in P to be r e d u n d a n t if there is an index j such that, for all x ~ P, x i ~ x w h e n e v e r x~ <~x for all i ~ j (which is equivalent to x i ~< ~/i*i x~ w h e n P is a join-semilattice). T h e breadth o[P, b(P), is defined as b e f o r e to be the sup of the lengths of the i r r e d u n d a n t sequences in P. T h e r e is also a dual definition of b r e a d t h f o r a poset P (wherein t> replaces ~<) which is clearly equal to b(P*). A s it turns out, however, the two definitions agree (cf. Ex. 3, p. 100 of [1]). 4.4. l ~ o p o s i t i o n . For any poset P, b(P) = b(P*). 1 ~ o o | . T h e statement that n <~b(P) is equivalent to the s t a t e m e n t that there exist x~ . . . . . x,, y~ . . . . . Yn in P such that, for all i and j, x i ~2, the m o n o i d g e n e r a t e d by A (see 3.2) is a 2 - p o s e t of
Cardinality questions concerning sernilattices
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infinite breadth, for if a, b are distinct elements of A , then, for every positive integer n, { a ~ b , a ~ - l b 2. . . . . ab"} is i r r e d u n d a n t (if x i = a " + l - ~ b i and p i = Xl...xj_lxj+l...x ., then x i ~ p i but x~<<-pi for i:~]). W e next p r o v e that every p o s e t P can be e m b e d d e d in a join-semilattice S of the s a m e breadth. M o r e o v e r , w h e n P is infinite and directed upward, S and its l o w e r sets h a v e the s a m e cardinality c h a r a c t e r as P. T h e definition of S is as follows. F o r any subset X of P, let X+={yeP:x<~y
for all x e X } ,
and let x + = {x} + for x e P. T h e n , X + = f"lx~x x+, X + N Y÷ = ( X t_J Y)+, a n d hence, S~+(p) = {F+: F e ~ ( P ) } is a meet-semilattice with respect to =_. T h e r e f o r e , S = S~+(P) is a join-semilattice with respect to __ and F + v G + = ( F t 3 G ) +. Since, for x, y e P, we clearly h a v e x ~< y if and only if x+__ y÷, the m a p x - - > x + is an o r d e r - i s o m o r p h i s m of P into S. M o r e o v e r , {x+: x e P} generates S since, f o r F = { x 1. . . . . x,}=_P, we have
i=l
i=l
T h e e m b e d d i n g x--->x + is, in fact, m o r e than just an o r d e r - i s o m o r p h i s m : it preserves sups. For, if A ~ P and p = s u p a ~ a a in P, then p + = f ' ) a ~ , a + and therefore, p+ = sup~,A a + in S. This proves the first part of the following proposition. 4.6. l~rOlmsilion. I f P is a poset, then {x+: x ~ P} generates the joiri-semilattice S = ~;+(P) a n d x ---->x + is an order-isomorphism o f P into S which preserves sups. In addition, (1) b ( S ) = b(P), (2) ISI = IP[, w h e n P is infinite, and (3) if P is directed upward, then for an infinite cardinal K, IP[x][ < K for all x ~ P if a n d only if ISEu]l < K for all u ~ S. Ibroot. (1) {x~}i"=1 is r e d u n d a n t in the poset P if and only if there is an index ] such that, for all x ~ P, xj <~ x w h e n e v e r xi ~< x for all i ~ j, i.e., x~ ~_ 0 ~ j x~- or, w h a t is the same, x~" ~< V ~ i x~- in the join-semilattice S. If we n o w identify P with {x+: x ~ P } , it follows f r o m the a b o v e and 4.3 that b ( P ) = b s ( P ) = b(S). (2) Since P is e m b e d d e d in S and F---> F + maps ~:(P) o n t o S, we have IPI ~< Isl ~
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5.5. Corollary. If ~; is a collection of finite sets and k is a positive integer, then I~1<~-1 when either of the following conditions holds: (1) ~; is closed under finite union and, for any k + 1 sets in ~;, one of the sets is contained in the union of the others. (2) ~: is closed under finite intersection and directed upward and, for any k + 1 sets in ~, one of the sets contains the intersection of the others. P r o o f . W h e n (1) holds, ~: is a lower finite join-semilattice with respect to ___ and b(~-) ~< k. W h e n (2) holds, then, by 3.6, ~ is a lower finite lattice with respect to ___ (since a collection ~0 of finite sets is directed u p w a r d if and only if it is cofinal in ~:(A), w h e r e A = U ~;0)- M o r e o v e r , as a meet-semilattice it has b r e a d t h ~< k so, by 4.4, b(~)<~k. H e n c e 5.5 follows f r o m 5.2. Suppose L is a lower finite lattice. T h e n , as noted earlier, L has a 0 and for any A ___L, we define the height of A, h ( A ) , to be the sup of the integers n I> 0 for which there exists a ~ A and a strictly increasing s e q u e n c e {x~}i%o in L with x o = 0 and x, = a. Since L is lower finite, each e l e m e n t x ~ L has a finite height h ( x ) - - h({x}). If A~ . . . . . A , are subsets of L, we define A ~ v . . . v A ~ to be the set of all a~ v - • • v a n, w h e r e a~ ~ A, for i = 1 . . . . . n, and w h e n each A i = A we writ~ A " for A~ v . • . v A ~ . If A v A has finite height w h e n e v e r A does, we shall say that join is bounded in L.
5.6. L e m m a . If A 1. . . . . A~ are subsets of finite height in a lower finite lattice L in which join is bounded, then A 1 v . • • v A . has finite height. lProo|. S u p p o s e n = 2 . T h e n B = A 1 U A 2 has finite height since h ( B ) = max{h(A1), h(A2) }. T h e r e f o r e , A l v A 2 has finite height since A ~ v A 2 ~ B v B and join is b o u n d e d in L. T h e result for general n follows easily by induction. 5.'/. q[laeorem. I f L is a lower finite lattice of finite breadth in which join is
bounded, then L is countable; in fact, for each n, the set of elements of L of height n is finite. l~root. L e t A d e n o t e the elements of L of height n and let {xi} be a s e q u e n c e in A . If b(L) = k, then { x l v - • • vxi} is an increasing s e q u e n c e in A k and, by 5.6, A k has finite height. T h e r e f o r e , for s o m e N, x l v . . . v x i = x l v . • - v x N for all i > N . T h e n , x~ ~
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shown [1, p. 40] that if a lower finite lattice L is u p p e r semimodular, then h ( x v y ) + h ( x ^ y ) ~ h ( x ) + h ( y ) for all x, y e L . Therefore, join is b o u n d e d in such a lattice since we have h(x x/y)~< h(x)+ h(y). This, together with 5.7, proves the following result.
If L is a lower finite lattice which is upper semimodular and has finite breadth, then L is countable. 5.8. Corollary.
6. Results and questions on existence H e r e we shall consider the following two questions: (A) For each positive integer k and ordinal h, does there exist a joinsemilattice having breadth k, lower index h, and cardinality Rx+k_l? (B) D o k-lattices of cardinality Nk_ 1 exist for every positive integer k? 6.1. For k = 1, the answer to both questions is yes: the ordinal Rx ()t = 0 for (B)) provides the required example. 6.2. For k = 2, the answer to (B) is also yes. T h e following r e m a r k s (the details of which are easily checked) show how a 2-lattice of cardinality R 1 can be constructed by transfinite induction. (We shall call a chain which is order-isomorphic to an ordinal a an a-chain.) (i) A countably infinite lattice L contains a cofinal to-chain C. For, if L = {x,: n ~ to}, then we can take
C={~/=oX,:neto ]. (ii) If L is a 2-lattice and C is a cofmal to-chain in L, then L can be extended to a 2-1attice L ' in which L is a p r o p e r ideal. T h e construction is simple. Let C' be a copy of C disjoint from L and let x ~ x' be an order-isomorphism of C onto C'. T h e n we take L ' = L U C', where the ordering of L ' extends both that of L and C' and is determined by the requirement that each x in C is covered by the corresponding x' in C' (i.e., in addition to the related pairs of L and C', we adjoin all pairs (a, b'), where a ~ L, b ~ C, and a ~< b in L). (iii) If {L~},~,A is a family of k-lattices which is nested (i.e., totally ordered by inclusion) and has the property that L~ is an ideal in L B whenever L~ ~ L B, and, if we set L = [ _ J ~ , L ~ and define x~I 3, we do not know the answer to (B). An approach which is parallel
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to the one outlined above, in (ii) would have L a k-lattice and C a cofinal ( k - 1)-lattice subset of L closed under ^ (by 3.4(2), the lattice requirement on C is automatically satisfied if C is cofinal in L and closed under ^). T h e p r o b l e m is showing such cofinal subsets exist at each stage of the construction. In this connection, suppose L o is a 2-lattice of cardinality R~, as constructed above, and L = L 0 x to with the product order (so L is a 3-lattice consisting of to copies of L 0 stacked one on top of the other). D o e s L contain a cofinal 2-lattice subset closed under ^ ? With respect to (B) then, the simplest open p r o b l e m is: P r o b l e m 1. D o e s there exist a 3-lattice of cardinality Rz? 6.4. For k = 2, we can show that the answer to (A) is yes when R~ is a regular cardinal. T h e method is very similar to the construction above. However, we need the following results to take the place of (ii) and (iii). (ii)' Let P be a join-semilattice, a an ordinal, and {I}v<~ a family of ideals in P such that P = (-J-r<~,Iv and Iv, _~Iv2 ff Vl
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P r o c e e d i n g by induction, we assume that /3 P b e a bijection and define {z.}.<.,, recursively as follows: z o e P is arbitrary and for 0 < ' r < t o x we set z~ = inf{z ~ C~: [('r) < z and z~ < z for all tr < "r}, w h e r e ~1 is the smallest ordinal exceeding zr(f('r)) and zr(z~) for all tr <-r. T h e n C = {z~}~<.,, is a cofinal tax-chain in P and we o r d e r S a = P U C a as described in (ii)'. (c) I f / 3 is a limit ordinal and cf(/3)< tax, then t h e r e exists {I~}~<~, a family of ideals in P, such that P=U~
x,o=x ~,
x~¢=irff{z ~Cv : x ~ , < z for all / z < ~ } .
T h e n {x~¢}¢<~ is increasing and we set I~ = U e < ¢ P [ x ~ ] . It is easy to verify that {I~}~<,,, has the p r o p e r t i e s stated above. ( W e need Rx regular to g u a r a n t e e that II~1<~.) W e o r d e r S a by attaching C a to P as in (ii)'. This c o m p l e t e s the construction of the family of join-semilattices {S~,}~,<. . . . . T h e r e q u i r e d join-semilattice S is Uv<~+, Sv.
Problem 2. D o e s there exist a join-semilattice of b r e a d t h 2, lower index to, and cardinality R,,+I? This is the simplest o p e n p r o b l e m with respect to (A).
References [1] G. Birkhoff, Lattice Theory, 3rd ed. (Amer. Math. Sac. Providence, RI, 1967). [2] P. ErdiSs and A. Hajnal, On the structure of set-mappings, Acta Math. (Hungary) 9 (1958) 111-131. [3] K. Kuratowski, Sur une caract6risation des alephs, Fund. Math. 38 (1951) 14,--17. [4] A. Rosenfeld, An Introduction to Algebraic Structures (Holden-Day, San Francisco, 1968).
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CARDINALITY QUESTIONS CONCERNING SEMILA.TrICES OF F I N r r E B R E A D T H S.Z. DITOR* Department of Mathematics, University of Western Ontar/o, London, Ontar/o, N 6 A 5B9, Canada
Received 22 September 1978 Revised 18 July 1983 It is shown that if L is a lattice in which every element has only finitely many predecessors and (*) every element has no more than k immediate predecessors, for some positive integer k, then I L l ~ k - r An example is constructed in which k = 2 and ILl=~t, but the question of whether ILl =~k-~ is possible for k > 2 is left unanswered. The conclusion ILl~
1. Introduction Let A be a set and ~:(A) the finite subsets of A ordered by inclusion. R. Haydon raised the following question concerning ~(A). D o e s t h e r e e x i s t a c o f i n a l s u b f a m i l y ~:0 (i.e., e v e r y G in ~ ( A ) is a s u b s e t o f s o m e F in ~ o ) w h i c h is c l o s e d u n d e r f i n i t e i n t e r s e c t i o n a n d is s u c h t h a t , f o r e v e r y F in ~ 0 , t h e c o l l e c t i o n o f p r o p e r s u b s e t s o f F w h i c h a r e in ~:o h a s at m o s t t w o maximal
members
(i.e., e v e r y
~:o-Set h a s
at m o s t
two maximal
proper
~0-
subsets)? Haydon
a s k e d this q u e s t i o n ( a n d a n s w e r e d it f o r t h e c a s e [A] ~
tion with a certain topology problem concerning the space of probability measures o n t h e p r o d u c t {0, 1} A, w h o s e r e p r e s e n t a t i o n as a n i n v e r s e l i m i t o v e r t h e i n d e x s e t ~:(A) offered an approach to solving the problem. When
A is c o u n t a b l e , t h e r e is n o difficulty in f i n d i n g s u c h a s u b f a m i l y ~ o o f
* This research was supported in part by a grant from the Natural Sciences and Engineering Research Council Canada. 0012-365X/84/$3,00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
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S.Z. Ditor
~ ( A ) . For, if A = { a 1, a 2. . . . }, we can take ~r0={F1, F 2. . . . }, where / 7 , = {a 1. . . . . a,}. It is when A is uncountable that the question becomes interesting. In this paper, we shall reformulate this question about ~r(A) in the setting of lattices. H e r e the essential elements of the p r o b l e m present themselves with greater clarity and several generalizations immediately b e c o m e apparent. A corollary of one of the theorems we p r o v e provides a s o m e w h a t surprising answer to the question raised by Haydon. W e show that if L is a lattice in which every element has only finitely m a n y predecessors and no m o r e than k immediate predecessors, for some positive integer k, then L has cardinality ILl~<~k_l. In addition, we provide an example in which k = 2 and ILl = R1. It follows f r o m these results that H a y d o n ' s subfamily ~:o exists if and only if IAI ~
2. Preliminaries Let us first recall a few definitions and introduce the terminology we shall be using. A poset (partially ordered set) is a set P equipped with a reflexive, antisymmetric, and transitive relation ~<. If, for all x, y in P, either x ~< y or y ~
y<~x for some x ~ X } ,
which is written P[x] when X = then X is said to be colinal in If y < x and for no z in P is the set of elements covered by
{x}. T h e upper set of X is P*[X]. W h e n P [ X ] = P, P. y < z < x, then we say that x covers y. W e denote x by cov(x):
coy(x) = {y: x covers y}. A poset P is said to be directed upward (resp. directed downward) if every pair of elements x, y in P has an upper bound (resp. lower bound); if there is a least upper bound, or sup, x v y (resp. greatest lower bound, or irff, x A y ) , then P is
Cardinality questions concerning semilattices
49
called a join-semilattice (resp. meet-semilattice). When both x v y and x ^ y exist for all x, y in P, then P is called a lattice. If P is a join-semilattice, then the operation x v y is associative, commutative, and idempotent and for any finite sequence x 1. . . . . x~ in P, the least upper bound sup{x1 . . . . . x,} exists and coincides with x l v - - - v x ~ , which we shall sometimes denote simply by V x~ (when the domain of i is understood). A similar statement holds for meet-semilattices. A subset M of a lattice P is called a sublattice if x v y and x ^ y are in M whenever x and y are in M, and a lattice subset if the poset M (with the ordering inherited from P) is a lattice. If P is a join-semilattice, then a subset I is called an ideal if x v y ~ I whenever x, y ~ L and P [ / ] = L W h e n P is a meet-semilattice, the condition P[I] = I is equivalent to r ^ x ~ 1 whenever r ~ P and x ~/. If a poset P has a smallest d e m e n t , we normally denote it by 0 and say that P has a 0. Of course, if P has a 0, then P is directed downward. Suppose, on the other hand, that the non-empty poset P is directed downward but has no 0. Then, given any xl ~ P, there exists y ~ P such that x~ ~ P [ y ] . If, now, x2 is a lower bound for {x~, y}, then we must have x 2 < xx. Continuing, we can next find x 3 < x 2, and so on. Hence, we see that a poset which is directed downward either has a 0 or contains a strictly decreasing infinite sequence.
3. Lower finite lattices and k-lattices W e shall call a poset P lower finite if P[x] is finite for all x in P. If P is lower finite and the function [cov(x)l is bounded on P with maximum value k, then we shall say that P is a k-poset. A k-poset which is also a lattice shall be called a k-lattice. If to is the set of natural numbers with their usual order, then, clearly, to is a I-lattice and to xto x - - . xto (k factors) with the product order ( ( n i ) ~ ( m l ) if and only if ni ~
S.Z. Direr
50
3.2. l~.Y~mple, d~(A), the m o n o i d g e n e r a t e d by the set A , consists of all finite sequences in A , including the e m p t y s e q u e n c e 0. T h e p r o d u c t of the two sequences s and t is the s e q u e n c e st o b t a i n e d by c o n c a t e n a t i o n , with ~ acting as an identity. If we define s ~< t to m e a n that t = usv for s o m e u a n d v, then J~t(A) b e c o m e s a lower finite poset which is directed u p w a r d a n d d o w n w a r d (¢ ~< s for all s). If a~ . . . . . a , e A and s = a 1 • • • a,, then cov(s) = {a 1 • • • a , _ 1, a 2 • • • a,}, so f o r IA[~>2, ~ ( A ) is a 2-poset. H o w e v e r , for IA[~>2, ~ t ( A ) is n o t a lattice since f o r distinct elements a, b in A , {ab, ba} has only ¢, a, and b as lower b o u n d s a n d therefore has no greatest lower b o u n d . W h e n A is infinite, [~t(A)l = IA[. 3.3. E x a m p l e . L e t S be the subfamily of ~ : ( A ) consisting of 0 a n d all the singletons {a}, a c A . T h e n S is a 1-poset which is a meet-semilattice a n d
ISI--IAl+l. T h e situations exemplified a b o v e c a n n o t be significantly e x p a n d e d b e c a u s e of the condition of lower finiteness. F o r example, if P is a lower finite joinsemilattice which is directed d o w n w a r d , then P is a lattice, since for any x, y in P, the set I of lower b o u n d s for {x, y} is a finite, n o n e m p t y ideal a n d t h e r e f o r e its least u p p e r b o u n d sup I exists and belongs t o / , i.e., x A y = sup L M o r e o v e r , by o u r earlier remarks, a n o n - e m p t y lower finite p o s e t is directed d o w n w a r d i~ and only if it has a 0. This proves the first part of the following proposition. 3.4. Proposition. (1) I f P is a lower finite ]oin-semilattice, then, with the ad]unction
of a 0 if necessary, P is a lattice. (2) I f P is a lower finite meet-semilattice which is directed upward, then P is a lattice. P r o o f . Suppose P x, y ~ P. W e must m e m b e r . Since S n o n - e m p t y subset l e m m a [4, p. 44].
is a lower finite meet-semilattice which is directed u p w a r d a n d show that S, the set of u p p e r b o u n d s for {x, y}, has a smallest is n o n - e m p t y a n d contains inf F = x~ ^ . • . ^ x , f o r every finite F = {x~ . . . . . x n} of S, t h e p r o o f of (2) follows f r o m the following
3.5. I~mmR. [_f p is a meet-semilattice in which every strictly decreasing sequence is finite and X is any non-empty subset of P, then inf X, the greatest lower bound of X, exists; moreover, inf X = xl A" • • A X, for some finite subset {x~ . . . . . x,} of X. l h ~ o L S u p p o s e the l e m m a is false. T h e n , for x~ . . . . . x r e X , x ~ ^ - • • ^ x r is not the greatest lower b o u n d of X, and so, t h e r e exists x , + ~ X such that x~ A" • • A Xr A Xr+~ < X1 A" • • ^ X,. It follows that S contains a strictly decreasing infinite sequence, c o n t r a r y to o u r hypothesis. T h e next proposition is n o w immediate.
Cardinality questions concerningsemilattices
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3.6. P r o p o s i t i o n . If ~;o is a cofinal subfamily of ~;( A ) and ~o is closed under finite
intersection, then ~:o is closed under arbitrary intersection and is a lower finite lattice with respect to inclusion. Conversely, if L is a lower finite lattice, then x --~ L[x] is an order-isomorphism (in fact, a meet-isomorphism) of L onto a cofinal subfamily of ~:(L) which is closed under intersection. L e t us return n o w to H a y d o n ' s question concerning ~r(A). If A is infinite and ~ o is cofinal in ~ ( A ) , then A = U ~0, f r o m which it is easily verified that [AI = I~r01. It t h e r e f o r e follows f r o m 3.6 and o u r previous r e m a r k s that if K ~>R 1, then H a y d o n ' s subfamily ~r0 exists for [A[ = K if and only if t h e r e exists a 2-lattice L with [ L I = K. T h e t h e o r e m that we shall p r o v e regarding k-lattices actually pertains to a larger class of semilattices, which we shall n o w consider.
4. Breadth In a join-semilattice S, a finite s e q u e n c e {x~}~=1 of length n is called redundant if t h e r e exists an index ] such that V x ~ = V ~ . j x ~ (equivalently, x ~ < V ~ . j x ~ ) ; otherwise, it is called irredundant. The sup of the lengths of the i r r e d u n d a n t sequences in S is called the breadth of S and is d e n o t e d by b(S). F o r e x a m p l e , if b ( S ) = 1, then S is a chain, since for all x, y in S, x v y is either x or y. 4.1. P r o p o s i t i o n . I l L is a k-lanice, then b(L)<~k. l~root. L e t L be a k-lattice and {x~}~__+~an arbitrary s e q u e n c e of length k + 1 in L. It suffices to show that V x~ = V~.ix~ for s o m e index j. L e t y = V x~, Yi = V~.jx~ (j = 1 . . . . . k + 1), and s u p p o s e that yj ~ y for all ]. T h e n Yi < Y for ] = 1 . . . . . k + 1 a n d for each j t h e r e exists z~ ~ cov(y) such that yj ~< z r Since Icov(y)l ~< k t h e r e exist distinct indices l, m such that zt = z m. H e n c e , yl v ym ~
52
S.Z. Ditor
T h e join-semilattice d e t e r m i n e d by S is o b t a i n e d by interpreting the p r o d u c t xy as the join operation, which a m o u n t s to defining x ~
Cardinality questions concerning sernilattices
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infinite breadth, for if a, b are distinct elements of A , then, for every positive integer n, { a ~ b , a ~ - l b 2. . . . . ab"} is i r r e d u n d a n t (if x i = a " + l - ~ b i and p i = Xl...xj_lxj+l...x ., then x i ~ p i but x~<<-pi for i:~]). W e next p r o v e that every p o s e t P can be e m b e d d e d in a join-semilattice S of the s a m e breadth. M o r e o v e r , w h e n P is infinite and directed upward, S and its l o w e r sets h a v e the s a m e cardinality c h a r a c t e r as P. T h e definition of S is as follows. F o r any subset X of P, let X+={yeP:x<~y
for all x e X } ,
and let x + = {x} + for x e P. T h e n , X + = f"lx~x x+, X + N Y÷ = ( X t_J Y)+, a n d hence, S~+(p) = {F+: F e ~ ( P ) } is a meet-semilattice with respect to =_. T h e r e f o r e , S = S~+(P) is a join-semilattice with respect to __ and F + v G + = ( F t 3 G ) +. Since, for x, y e P, we clearly h a v e x ~< y if and only if x+__ y÷, the m a p x - - > x + is an o r d e r - i s o m o r p h i s m of P into S. M o r e o v e r , {x+: x e P} generates S since, f o r F = { x 1. . . . . x,}=_P, we have
i=l
i=l
T h e e m b e d d i n g x--->x + is, in fact, m o r e than just an o r d e r - i s o m o r p h i s m : it preserves sups. For, if A ~ P and p = s u p a ~ a a in P, then p + = f ' ) a ~ , a + and therefore, p+ = sup~,A a + in S. This proves the first part of the following proposition. 4.6. l~rOlmsilion. I f P is a poset, then {x+: x ~ P} generates the joiri-semilattice S = ~;+(P) a n d x ---->x + is an order-isomorphism o f P into S which preserves sups. In addition, (1) b ( S ) = b(P), (2) ISI = IP[, w h e n P is infinite, and (3) if P is directed upward, then for an infinite cardinal K, IP[x][ < K for all x ~ P if a n d only if ISEu]l < K for all u ~ S. Ibroot. (1) {x~}i"=1 is r e d u n d a n t in the poset P if and only if there is an index ] such that, for all x ~ P, xj <~ x w h e n e v e r xi ~< x for all i ~ j, i.e., x~ ~_ 0 ~ j x~- or, w h a t is the same, x~" ~< V ~ i x~- in the join-semilattice S. If we n o w identify P with {x+: x ~ P } , it follows f r o m the a b o v e and 4.3 that b ( P ) = b s ( P ) = b(S). (2) Since P is e m b e d d e d in S and F---> F + maps ~:(P) o n t o S, we have IPI ~< Isl ~
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5.5. Corollary. If ~; is a collection of finite sets and k is a positive integer, then I~1<~-1 when either of the following conditions holds: (1) ~; is closed under finite union and, for any k + 1 sets in ~;, one of the sets is contained in the union of the others. (2) ~: is closed under finite intersection and directed upward and, for any k + 1 sets in ~, one of the sets contains the intersection of the others. P r o o f . W h e n (1) holds, ~: is a lower finite join-semilattice with respect to ___ and b(~-) ~< k. W h e n (2) holds, then, by 3.6, ~ is a lower finite lattice with respect to ___ (since a collection ~0 of finite sets is directed u p w a r d if and only if it is cofinal in ~:(A), w h e r e A = U ~;0)- M o r e o v e r , as a meet-semilattice it has b r e a d t h ~< k so, by 4.4, b(~)<~k. H e n c e 5.5 follows f r o m 5.2. Suppose L is a lower finite lattice. T h e n , as noted earlier, L has a 0 and for any A ___L, we define the height of A, h ( A ) , to be the sup of the integers n I> 0 for which there exists a ~ A and a strictly increasing s e q u e n c e {x~}i%o in L with x o = 0 and x, = a. Since L is lower finite, each e l e m e n t x ~ L has a finite height h ( x ) - - h({x}). If A~ . . . . . A , are subsets of L, we define A ~ v . . . v A ~ to be the set of all a~ v - • • v a n, w h e r e a~ ~ A, for i = 1 . . . . . n, and w h e n each A i = A we writ~ A " for A~ v . • . v A ~ . If A v A has finite height w h e n e v e r A does, we shall say that join is bounded in L.
5.6. L e m m a . If A 1. . . . . A~ are subsets of finite height in a lower finite lattice L in which join is bounded, then A 1 v . • • v A . has finite height. lProo|. S u p p o s e n = 2 . T h e n B = A 1 U A 2 has finite height since h ( B ) = max{h(A1), h(A2) }. T h e r e f o r e , A l v A 2 has finite height since A ~ v A 2 ~ B v B and join is b o u n d e d in L. T h e result for general n follows easily by induction. 5.'/. q[laeorem. I f L is a lower finite lattice of finite breadth in which join is
bounded, then L is countable; in fact, for each n, the set of elements of L of height n is finite. l~root. L e t A d e n o t e the elements of L of height n and let {xi} be a s e q u e n c e in A . If b(L) = k, then { x l v - • • vxi} is an increasing s e q u e n c e in A k and, by 5.6, A k has finite height. T h e r e f o r e , for s o m e N, x l v . . . v x i = x l v . • - v x N for all i > N . T h e n , x~ ~
Cardinalityquestionsconcerningsemilattices
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shown [1, p. 40] that if a lower finite lattice L is u p p e r semimodular, then h ( x v y ) + h ( x ^ y ) ~ h ( x ) + h ( y ) for all x, y e L . Therefore, join is b o u n d e d in such a lattice since we have h(x x/y)~< h(x)+ h(y). This, together with 5.7, proves the following result.
If L is a lower finite lattice which is upper semimodular and has finite breadth, then L is countable. 5.8. Corollary.
6. Results and questions on existence H e r e we shall consider the following two questions: (A) For each positive integer k and ordinal h, does there exist a joinsemilattice having breadth k, lower index h, and cardinality Rx+k_l? (B) D o k-lattices of cardinality Nk_ 1 exist for every positive integer k? 6.1. For k = 1, the answer to both questions is yes: the ordinal Rx ()t = 0 for (B)) provides the required example. 6.2. For k = 2, the answer to (B) is also yes. T h e following r e m a r k s (the details of which are easily checked) show how a 2-lattice of cardinality R 1 can be constructed by transfinite induction. (We shall call a chain which is order-isomorphic to an ordinal a an a-chain.) (i) A countably infinite lattice L contains a cofinal to-chain C. For, if L = {x,: n ~ to}, then we can take
C={~/=oX,:neto ]. (ii) If L is a 2-lattice and C is a cofmal to-chain in L, then L can be extended to a 2-1attice L ' in which L is a p r o p e r ideal. T h e construction is simple. Let C' be a copy of C disjoint from L and let x ~ x' be an order-isomorphism of C onto C'. T h e n we take L ' = L U C', where the ordering of L ' extends both that of L and C' and is determined by the requirement that each x in C is covered by the corresponding x' in C' (i.e., in addition to the related pairs of L and C', we adjoin all pairs (a, b'), where a ~ L, b ~ C, and a ~< b in L). (iii) If {L~},~,A is a family of k-lattices which is nested (i.e., totally ordered by inclusion) and has the property that L~ is an ideal in L B whenever L~ ~ L B, and, if we set L = [ _ J ~ , L ~ and define x~
58
S.Z.
Ditor
to the one outlined above, in (ii) would have L a k-lattice and C a cofinal ( k - 1)-lattice subset of L closed under ^ (by 3.4(2), the lattice requirement on C is automatically satisfied if C is cofinal in L and closed under ^). T h e p r o b l e m is showing such cofinal subsets exist at each stage of the construction. In this connection, suppose L o is a 2-lattice of cardinality R~, as constructed above, and L = L 0 x to with the product order (so L is a 3-lattice consisting of to copies of L 0 stacked one on top of the other). D o e s L contain a cofinal 2-lattice subset closed under ^ ? With respect to (B) then, the simplest open p r o b l e m is: P r o b l e m 1. D o e s there exist a 3-lattice of cardinality Rz? 6.4. For k = 2, we can show that the answer to (A) is yes when R~ is a regular cardinal. T h e method is very similar to the construction above. However, we need the following results to take the place of (ii) and (iii). (ii)' Let P be a join-semilattice, a an ordinal, and {I}v<~ a family of ideals in P such that P = (-J-r<~,Iv and Iv, _~Iv2 ff Vl
Cardinality questions concerning semilattices
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P r o c e e d i n g by induction, we assume that /3
x,o=x ~,
x~¢=irff{z ~Cv : x ~ , < z for all / z < ~ } .
T h e n {x~¢}¢<~ is increasing and we set I~ = U e < ¢ P [ x ~ ] . It is easy to verify that {I~}~<,,, has the p r o p e r t i e s stated above. ( W e need Rx regular to g u a r a n t e e that II~1<~.) W e o r d e r S a by attaching C a to P as in (ii)'. This c o m p l e t e s the construction of the family of join-semilattices {S~,}~,<. . . . . T h e r e q u i r e d join-semilattice S is Uv<~+, Sv.
Problem 2. D o e s there exist a join-semilattice of b r e a d t h 2, lower index to, and cardinality R,,+I? This is the simplest o p e n p r o b l e m with respect to (A).
References [1] G. Birkhoff, Lattice Theory, 3rd ed. (Amer. Math. Sac. Providence, RI, 1967). [2] P. ErdiSs and A. Hajnal, On the structure of set-mappings, Acta Math. (Hungary) 9 (1958) 111-131. [3] K. Kuratowski, Sur une caract6risation des alephs, Fund. Math. 38 (1951) 14,--17. [4] A. Rosenfeld, An Introduction to Algebraic Structures (Holden-Day, San Francisco, 1968).