Sep 29, 1978 - Abstract. This paper defines the anti-exchange closure, a generalization of the order ideals of a partially ordered set. Various theorems are ...
Birkh~iuser Verlag, Basel
Algebra Universalis, 10 (1980) 290-299
M e e t - d i s t r i b u t i v e lattices and the a n t i - e x c h a n g e closure PAUL H. EDELMAN*
Abstract. This paper defines the anti-exchange closure, a generalization of the order ideals of a partially ordered set. Various theorems are proved about this closure. The main theorem presented is that a lattice L is the lattice of closed sets of an anti-exchange closure if and only if it is a meet-distributive lattice. This result is used to give a combinatorial interpretation of the zetapolynomial of a meet-distributive lattice.
1. Introduction G i v e n a finite set S, a f u n c t i o n ~b which m a p s the set of all s u b s e t s of S, P ( S ) , to P(S) is called a closure r e l a t i o n if it satisfies the p r o p e r t i e s 1) ~b(A) = &2(A) 2) A c 4,(A) 3) if A c B
t h e n d~(A)c~(B).
T h o s e subsets in the r a n g e of 4) are called the closed sets. F o r t h e p u r p o s e of simplicity we a s s u m e that the null set is closed. 4~(A) is m o r e c o m m o n l y d e n o t e d ,~ w h e n t h e r e can be n o a m b i g u i t y . A l t e r n a t i v e l y , given a collection cr of subsets of S, the null set a n d S b o t h in cr such that if A ~ qr a n d B s cr t h e n A n / 3 ~ cr t h e n .cr is the collection of closed sets for s o m e closure with
6(A)= n D, DiE-C A=D~
A closure r e l a t i o n o n a set can p r o d u c e q u i t e a rich s t r u c t u r e . A m o n g the m o s t
* Work done while the author was an Applied Mathematics Fellow at M.I.T.. Presented by R. P. Dilworth. Received March 7, 1978. Accepted for publication in final form September 29, 1978. 290
Vol. 10, 1 9 8 0
Meet-distributive lattices and the anti-exchange closure
291
extensively studied finite closures is the exchange closure. This closure satisfies the additional property that given a closed set A and two elements x and y, neither in A, then y ~ A U x implies x ~ A U y. Sets with this closure are called combinatorial geometries or matroids. (See [2].) A n o t h e r type of finite closure relation can be defined on a partially o r d e r e d set P. If A is a subset of P then define ~b(A) = {p I P ~< a for some a ~ A}. U n d e r this closure the closed sets are the order ideals of A and hence we will call this the order ideal closure. By a well-known theorem of Birkhoff [1] a lattice L is distributive if and only if it is the lattice of order ideals, ordered by inclusion, of some poset P. We define J(P) to be the lattice of order ideals of P and thus have that L is distributive if and only if it is isomorphic to J(P) for some P. In this p a p e r we present a generalization of the order ideal closure. "ilais closure satisfies the anti-exchange property, that is given a closed set A and two elements x and y, not identical and not in A, then x~AUy
implies
y~AUx.
The main theorem is that a lattice L is the lattice of closed sets of an antiexchange closure if and only if it is meet-distributive. Section II of this p a p e r introduces the anti-exchange closure, presents some examples and proves some elementary properties. In section I I I we p r o v e the main t h e o r e m relating anti-exchange closures and meet-distributive lattices, and in section I V we present an application of this characterization to zeta polynomials.
2. The anti-exchange closure In this section we define the anti-exchange closure, present s o m e examples, and prove some elementary properties. Let S be a finite set and 4' a closure relation. Then we say 4' is anti-exchange if, given a closed set A, and two unequal elements of S, x and y, neither in A, then xeAOy
implies
yCAOx
This is called the anti-exchange closure since it has the p r o p e r t y opposite to that of the exchange closure.
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p . H . EDELMAN
ALGEBRA UNIV.
E X A M P L E 1. T h e order ideal closure of a poset is anti-exchange. E X A M P L E 2. L e t P be a poset and A a subset of P. Define ft, = { p e P ]
a~~0
Z ( n ) = the n u m b e r of o r d e r - p r e s e r v i n g m a p s o" : P--~ I n ] ( - 1 ) k Z ( - n ) = the n u m b e r of strict o r d e r - p r e s e r v i n g m a p s , ~- : P ~ I n ]
(Remark: a map o" is order-preserving if x
