Congruence lattices of pseudocomplemented semilattices

Congruence lattices of algebras in various varieties have been studied extensively in the literature. For example, congruence lattices (i.e. lattices of ideals) of Boolean algebras were characterized by Nachbin [18] (see also Gratzer [9] and Jonsson [16]) while congruence lattices of semilattices were investigated by Papert [19], Dean and Oehmke [4] and others. In this paper we initiate the investigations into the structure of congruence lattices of pseudocomplemented semilattices. Our main concern here is two-fold: to develop certain indispensable tools, such as filter congruences, for the study of congruence lattices and to show that they possess several interesting (special) properties in addition to being algebraic. After introducing the class of pseudocomplemented semilattices as a variety in Section 1, a very special congruence ~ is defined. In Section 2 certain congruences, called filter congruences, are defined and then used to express every congruence as a join of two "simpler" congruences (in fact even better, see Theorem 2.6), a decomposition which is important for our subsequent investigations. In Section 3 we analyse the possibility of dually embedding a given pseudocomplemented semilattice in its congruence lattice, while Section 4 deals with several special properties of congruence lattices. Most of the results in this paper form a part of the author's Ph.D. Thesis which was submitted to the University of Waterloo in 1974 and was written under the direction of Professor Stanley Burris. The author expresses his deep indebtedness and high appreciation to Professor Burris for his keen interest, constructive criticism and valuable remarks. The author is also grateful to Dr. S. BulmanFleming for several helpful conversations and for his interest in this paper. This work was essentially supported by an Ontario Graduate Fellowship and the final draft was prepared during January 1977 when the author was visiting the Department of Pure Mathematics, University of Waterloo and was supported by NRC Grant A7256.