Conjunctive Join Semi-Lattices

Abstract

A join-semilattice $L$ is said to be conjunctive if it has a top element $1$ and it satisfies the following first-order condition: for any two distinct $a,b\in L$, there is $c\in L$ such that either $a\vee c\not=1=b\vee c$ or $a\vee c=1\not=b\vee c$. Equivalently, a join-semilattice is conjunctive if every principal ideal is an intersection of maximal ideals. We present simple examples showing that a conjunctive join-semilattice may fail to have any prime ideals. (Maximal ideals of a join-semilattice need not be prime.) We show that every conjunctive join-semilattice is isomorphic to a join-closed subbase for a compact $T_1$-topology on $\mathrm{Max} L$, the set of maximal ideals of $L$. The representation is canonical in that when applied to a join-closed subbase for a compact $T_1$-space $X$, the space produced by the representation is homeomorphic with $X$. We say a join-semilattice morphism $\phi:L\to M$ is conjunctive if $\phi^{-1}(w)$ is an intersection of maximal ideals of $L$ whenever $w$ is a maximal ideal of $M$. We show that every conjunctive morphism between conjunctive join-semilattices is induced by a multi-valued function from $\mathrm{Max} M$ to $\mathrm{Max} L$. A base for a topological space is said to be annular if it is a lattice, and Wallman if it is annular and for any point $u$ in any basic open $U$, there a basic open $V$ that misses $u$ and together with $U$ covers $X$. It is easy to show that every Wallman base is conjunctive. We give an example of a conjunctive annular base that is not Wallman. Finally, we examine the free distributive lattice $dL$ over a conjunctive join semilattice $L$. In general, it is not conjunctive, but we show that a certain canonical, algebraically-defined quotient of $dL$ is isomorphic to the sub-lattice of the topology of the representation space that is generated by $L$. We describe numerous applications.