On the congruence lattice of a semilattice

Here is attempted an examination of three aspects of the lattice theta of congruence relations of a semilattice S (usually a join semilattice). The impetus for two of the three investigations is provided by the recent result of Fajtlowicz and Schmidt [5] that theta is dual to the lattice [theta](S) of algebraic closure subfamilies of S (the ideal lattice of S, [empty set] included). So in Chapter I we generalize the notion of algebraic closure family and examine the lattice [theta](X) of algebraic closure families in a complete lattice X. There we obtain a second order characterization theorem for [theta](X) by axiomatizing the occurence in theta of a copy of X[raised op] (the dual of X). This is called the upper spot in theta. Of course a characterization of [theta](S) (and so of theta) follows in the case that X is algebraic. We also find that the posession by a complete atomic lattice L of such a spot is tantamount to its being decomposable in a certain nice way ("atomwise") into disjoint complete join semilattices. Chapter II plays off the fact that [theta](S) is atomic and investigates the possibility that it is the presence (in sufficient quantities) of certain special types of atoms (especially primes) which results in [theta](S)'s various properties. The notion of a prime atom is presented in [theta](S) as a correlate to the notion of finitely meet irreducible element of S and it is shown that under certain conditions a complete atomic lattice with enough prime atoms will be dually-semi-Brouwerian, M*-symmetric (so lower semimodular) and, if algebraic, fully dually quasi-decomposable. So [theta](S) gets these properties, hence theta their duals. In Chapter III we take a quite different tack. Here only, we work with meet semilattices S and [theta](S) does not come into play. Our work is based on the fact that for a meet semilattice M, theta is distributive iff M is a tree. So a concrete examination of semilattice trees and their congruences is attempted in the first part of this chapter. But gradually, we find that the compact congruences on M form a Boolean ring, and a special one at that. We then find ourselves studying the Boolean ring B[M] universal over a meet semilattice M. In the end we find that M is a semilattice tree iff theta [congruent to] [cursive I](E[lowered M]) the ideal lattice of the evenly generated ideal of B[M]. So with E[lowered M] a Boolean ring we see that the compact congruences of a tree T form a Boolean ring E[lowered T]. Better yet if T has a zero, E[lowered T] = B[T] so theta [congruent to] cursive I so compact(theta) [congruent to] B[T].