ℵ₀-categorical distributive lattices of finite breadth

Every N „-categorical distributive lattice of finite breadth has a finitely axiomatizablc theory. This result extends the analogous result for partially ordered sets of finite width. 0. Introduction. This note is mainly concerned with the following theorem. Theorem 1. Every S „-categorical distributive lattice of finite breadth has a finitely axiomatizable theory. For general references on distributive lattice theory we suggest [1 and 5]. Relevant material concerning N„-categoricity can be found in [8 and 9]. The article [7] also considers S „-categorical distributive lattices, but in a slightly different vein than is done here. Theorem 1 generalizes a corresponding result for partially ordered sets (Theorem 1 of [9]) that every S „-categorical poset of finite width has a finitely axiomatizable theory. In fact, this latter result, which is an important ingredient in the proof of Theorem 1, follows fairly immediately from Theorem 1 as will be demonstrated in §3 by appropriately interpreting the theory of posets of width n in the theory of distributive lattices of breadth n. The notion of breadth of a lattice seems not to be a well discussed concept; it is not mentioned in either of the references [1 or 5], but is briefly referred to in [2]. Yet. for distributive lattices it becomes a particularly stable and transparent notion. Various equivalent characterizations will be presented in §1. The proof of Theorem 1 will be presented in §2. 1. Breadth. A lattice (A, A, V) has breadth < n iff whenever a0, a,.a„ E A there is some ; ^ n such that a0 V a, V • • • Van = a„ V a, V ■ ■ ■ VúiH V al+, V • • Van. If one identifies (as we shall do) the lattice (A, A, V) with the poset (A, <) (where x ^ y iff x = x A y ifíy = x V y), then it is immediate that breadth 1 lattices are merely chains. If L0, L,_.L„_, are chains, then L„ X L, X • • • XLB_, is a distributive lattice which has breadth «s n, and if each L, is nontrivial (i.e. \Lj\>2), then it has breadth precisely n. In particular, if each | L, |= 2, then Bn = L0 X • •■ X L„_, is the Boolean lattice with breadth n.