Relative Ockham lattices: their order-theoretic and algebraic characterisation

Abstract

1. Introduction. Given a variety si of lattice-ordered algebras, a lattice L is said to be a relative si-lattice if every closed interval [a, b] of L may be given the structure of an algebra in si (in other words, is the reduct of a member of si-not necessarily unique). This paper discusses the characterisation in terms of forbidden substructures of finite relative .stf-lattices. We treat a large class of varieties si of distributive-lattice-ordered algebras. For these varieties, the finite algebras can be described dually in terms of finite ordered sets, so that order-theoretic results and techniques prove valuable.Our study was prompted by a desire to set in a wider context the following characterisations of relative de Morgan lattices and of relative Stone lattices.

A note on essentially convex orders

“…It has been possible to characterize finite boolean, relative de Morgan, relative Stone and relative dual Stone lattices, by means of forbidden intervals, which turn out to be the smallest lattices that are not boolean, de Morgan, Stone or dual Stone, respectively (see [1,4,3]). The relationship between closed intervals in a distributive lattice and convex subposets of a poset, is described in the following Lemma 1.1 (G. Bordalo and H. Priestley, [3]).…”

“…The relationship between closed intervals in a distributive lattice and convex subposets of a poset, is described in the following Lemma 1.1 (G. Bordalo and H. Priestley, [3]). Let L be a finite distributive lattice, J(L) the set of its join-irreducible elements with the partial order induced from L. We recall that a subset P of a poset (A, ≤) is convex if c ∈ A, a, b ∈ P , a < c < b imply c ∈ P .…”

Decompositions for Relatively Normal Lattices

“…(1) =*• (2) Suppose that T = z° V z1 V • • • V zN, where zJ = VB (7). the sets B(j) are as described in Construction 4.7 for 0 < j < N, and B(0) is a maximal orthogonal set of maximal linear elements.…”

“…Some relevant publications include Beazer [4], Bórdalo [6], Bórdalo and Priestly [7], Cornish [13,14], Davey [16], Johnstone [19], Mandelker [20], Monteiro [24,25], Snodgrass and Tsinakis [26,27], and Zaanen [28]. The reader is advised that some of the cited authors express normality as a property of the lattice of closed sets of a topological space.…”

Rees congruences in lattice-ordered algebras

“…We identify L with O(P ). We invoke Lemma 3.2 of [1], which establishes that, under duality, the correspondence J → J (J) sets up a bijection between intervals J in L and convex subposets of P . Given a convex subposet Q of P , the associated interval [u, v]…”