Conjunctive join-semilattices

Abstract

A join-semilattice L with top is said to be conjunctive if every principal ideal is an intersection of maximal ideals. (This is equivalent to a first-order condition in the language of semilattices.) In this paper, we explore the consequences of the conjunctivity hypothesis for L, and we define and study a related property, called “ideal conjunctivity,” which is applicable to join-semilattices without top. Results include the following: (a) Every conjunctive join-semilattice is isomorphic to a join-closed subbase for a compact $T_1$-topology on $\mathop {\mathrm {max}}L$, the set of maximal ideals of L, and under weak hypotheses this representation is functorial. (b) Every Wallman base for a topological space is conjunctive; we give an example of a conjunctive annular base that is not Wallman. (c) The free distributive lattice over a conjunctive join-semilattice L is a subsemilattice of the power set of $\mathop {\mathrm {max}}L$. (d) For an arbitrary join-semilattice L: if every u-maximal ideal is prime (i.e., the complement is a filter) for every $u\in L$, then L satisfies Katriňák’s distributivity axiom. (This appears to be new, though the converse is well known.) If L is conjunctive, all the 1-maximal ideals of L are prime if and only if L satisfies a weak distributivity axiom due to Varlet. We include a number of applications.