Prime ideal characterization of chain based lattices

Abstract

Epstein and Horn, in their paper 'Chain based lattices', characterized P, -lattices, and P 2 -lattices in terms of their prime ideals. But no such prime ideal characterization for P 0 -lattices was given. Our main aim in this paper is to characterize P 0 -lattices in terms of their prime ideals. We also give a necessary and sufficient condition for a P-algebra to be a P 0 -lattice (and hence a /Vlattice). Keywords and phrases : B-normal lattices, P-algebra, chain based lattices, sheaf of bounded distributive lattices, continuity axiom, s^{L) is determined by a subset of L. IntroductionTraczyk (1963) introduced the concept of chain based lattices (P 0 -lattices) as an abstraction of Post algebras. Epstein and Horn (1975) studied chain based lattices in detail and obtained a prime ideal characterization of these lattices in special cases, namely, /Vlattices and P 2 -lattices. But no such prime ideal characterization for P olattices was given. In this paper, we give a prime ideal characterization for P olattices. The main tool used in this paper is that every bounded distributive lattice is isomorphic with the lattice of all global sections of a sheaf of bounded distributive lattices over a Boolean space (See Maddana Swamy (1974) and Subrahmanyam (1978)).Epstein and Horn (1975) (Theorem 7.3) proved that the prime ideals of a P 0 -lattice of order n lie in disjoint maximal chains each with at most n -1 elements and they have shown that the converse is not true even in the case of a P-algebra. In this paper, we prove that a P-algebra L is a P 0 -lattice (and hence a P 2 -lattice) if and only if the prime ideals of L lie in disjoint maximal chains each with at most n -1 elements for some integer n and satisfy the continuity axiom (see Definition 3.3 and Theorem 3.5 use, available at https://www.cambridge.org/core/terms. https://doi