Schmidt, Jurgen2021-12-172021-12-1719752669233https://hdl.handle.net/10657/8366Here is attempted an examination of three aspects of the lattice [theta](S) 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](S) 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](X) of a copy of X[raised op] (the dual of X). This is called the upper spot in [theta](X). Of course a characterization of [theta]*(S*) (and so of [theta](S)) 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](S) 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](M) 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](M) [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](T) [congruent to] [cursive I](B[T]) so compact([theta](T)) [congruent to] B[T].application/pdfenThis item is protected by copyright but is made available here under a claim of fair use (17 U.S.C. §107) for non-profit research and educational purposes. Users of this work assume the responsibility for determining copyright status prior to reusing, publishing, or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires express permission of the copyright holder.Thesisreformatted digital