A theorem on infinite distributivity for postL-algebras

Abstract

It is known that every Boolean algebra B satisfies the infinite distributive laws while an arbitrary distributive lattice may fail to satisfy either or both of these laws. In this note we show that both (DO and (Dz) hold for a Post L-algebra P= (B, L) with a finite lattice of constants L. Post L-algebras were introduced by T. P. Speed in [2] and further investigated by the author in [3] and [4]. It is shown in [2] that if L is a bounded distributive lattice, then every Post L-algebra P is isomorphic to the coproduct of a Boolean algebra B and L, where the coproduct is taken in the category of bounded distributive lattices and lattice homomorphisms preserving 0 and 1. P will be denoted by P= (B, L).All lattices considered in this note will be distributive lattices with 0 and 1, and all lattice homomorphisms will preserve 0 and 1. We shall use the terminology and notation of [1]. In particular, if L' is a sublattice of L and SC=L ", then the least L' upper bound of S in L' and L will be denoted (whenever they exist) by Z x and xES L Z x respectively, similar notation will be used for the greatest lower bounds of S xES in L' and L. We recall the definition of coproduct.DEFINrrIoN. Let L1, L2 and L be distributive lattices and let ix: Lx-~L and i,,: Ls-~ L be lattice monomorphisms. ~Ihe pair (L, {iI, is}) will be called the coproduet (=free product) of L1 and Ls ff for every distributive lattice D and every pair of lattice homomorphisms hi:/--1 ~ D and hs: Ls-~D, there is a unique lattice homomorphism h: L~ D such that hi~ = h~ and his = hs.We shall denote the coproduct of L~ and Ls by/,1. Ls and to simplify the notation we shall identify Lx and L2 with their isomorphic images ix(/-a) and is (La)

Coproducts of bounded distributive lattices: infinite distributivity

“…In [12], Speed introduces generalized Post algebras. A generalized Post algebra is a coproduct of a Boolean lattice B with an arbitrary bounded distributive lattice L. In [14], Yaqub shows that, if L is finite, then the generalized Post algebra satifies JID, and states that, in general, it is not known whether JID holds in the coproduct B * L. (Of course a necessary condition is that L itself must satisfy JID.) Our result answers this question.…”