Nearly twenty years ago, two of the authors wrote a paper on congruence lattices of semilattices [9]. The problem of finding a really useful characterization of congruence lattices of finite semilattices seemed too hard for us, so we went on to other things. Thus when Steve Seif asked one of us at the October 1990 meeting of the AMS in Amherst what we had learned in the meantime, the answer was...

The lattice Ì of lambda theories is isomorphic to the congruence lattice of the term algebra of the minimal lambda theory ¬. This remark is the starting point for studying the structure of Ì by universal algebraic methods. In this paper we show that Ì satisfies nontrivial lattice quasi-identities, while we conjecture that Ì does not satisfy any nontrivial lattice identity (a lattice identity is...

We define a G-fuzzy congruence, which is a generalized fuzzy congruence, discuss some of its basic properties, and characterize the G-fuzzy congruence generated by a fuzzy relation on a semigroup. We also give certain lattice theoretic properties of Gfuzzy congruences on semigroups.

We show that a locally finite variety is congruence join semidistributive if and only if it satisfies a congruence identity that is strong enough to force join semidistributivity in any lattice.

In the present note we characterize finite lattices which are isomorphic to the congruence lattice of an abelian lattice ordered group.

We prove that every finite distributive lattice D can be represented as the congruence lattice of a rectangular lattice K in which all congruences are principal. We verify this result in a stronger form as an extension theorem.

We redefine a fuzzy congruence, discuss some properties of the fuzzy congruences, find the fuzzy congruence generated by a fuzzy relation on a semigroup, and give some lattice theoretic properties of the fuzzy congruences on semigroups.

We give a necessary and su¢ cient condition for a bounded involution lattice to be isomorphic to the direct square of its invariant part. This result is applied to show relations between related lattices of an algebra: For instance, generalizing some earlier results of G. Czédli and L. Szabó it is proved that any algebra admits a connected compatible partial order whenever its quasiorder lattic...

The aim of the paper is to investigate some local properties of the weak congruence lattice of an algebra, which is supposed to possess the constant 0, or a nullary term operation. Lattice identities are restricted to the zero blocks of weak congruences. In this way, a local version of the CEP, and local modularity and distributivity of the weak congruence lattices are characterized. In additio...