Mildly distributive semilattices

Abstract

There is no single generalization of distributivity to semilattices. This paper investigates the class of mildly distributive semilattices, which lies between the two most commonly discussed classes in this area-weakly distributive semilattices and distributive semilattices. Particular attention is paid to describing and characterizing congruence distributive mildly distributive semilattices, in contrast to distributive semilattices, whose lattice of join partial congruences is badly behaved and which are difficult to describe.

A topological duality for mildly distributive meet-semilattices

“…The algebraic concepts on mildly distributive meet-semilattices presented in this section are due to Hickman [12] and due to us [4]; we direct the reader to these references for more details.…”

“…(2) If a b, then there exists P ∈ Opt(L) such that a ∈ P and b / ∈ P . is a strong homomorphism (see [12] and [2]) if it is a homomorphism and for all a 1 , . .…”

“…If h : L → M is a strong homomorphism, then h is a join-homomorphism and the converse is not always true. But, if L is an md-semilattice and M is an arbitrary meet-semilattice, then a map h : L → M is a strong homomorphism iff it is a join-homomorphism (this can be seen in [12]). Thus, strong homomorphisms and join-homomorphisms coincide for mdsemilattices.…”

“…The purpose of this section is to represent topologically some congruences on md-semilattices. Certain congruence relations on md-semilattices were studied and described by Hickman in [12]. Here we focus on a subclass of those congruences considered by Hickman.…”

On distributive join-semilattices

“…Different notions of distributivity for semilattices have been proposed in the literature as a generalization of the usual distributive property in lattices. As far as we know, notions of distributivity for semilattices have been given, in chronological order, by Grätzer and Schmidt [8] in 1962, by Katriňák [11] in 1968, by Balbes [1] in 1969, by Schein [14] in 1972, by Hickman [10] in 1984, and by Larmerová and Rachůnek [13] in 1988. Following the names of its authors, we will use the terminology GS-, K-, B-, S n -, H-, and LRdistributivity, respectively.…”