A topological duality for mildly distributive meet-semilattices

Abstract

We develop a topological duality for the category of mildly distributive meet-semilattices with a top element and certain morphisms between them. Then, we use this duality to characterize topologically the lattices of Frink ideals and filters, and we also obtain a topological representation for some congruences on mildly distributive meet-semilattices.

On the implicative‐infimum subreducts of weak Heyting algebras

“…The following lemma is a useful characterization of irreducible filters [7]. Lemma Let $$A$$ be a semilattice and $$F$$ a proper filter of $$A$$.…”

“…The following characterization is quite useful in practice (cf. [7]). Lemma Let $$A$$ be a semilattice and $$F$$ a proper filter of $$A$$.…”

“…There are different generalizations of the notion of distributivity from lattices to semilattices. In this paper we say that a semilattice $$A$$ is distributive [7] if for every $$a,b,c\in A$$ such that $$a\wedge b\le c$$ there exist $$a^{\prime }$$, $$b^{\prime }$$ in $$A$$ such that $$a\le a^{\prime }$$, $$b\le b^{\prime }$$, and $$a^{\prime }\wedge b^{\prime }=c$$. If $$A$$ is a distributive semilattice then every prime filter of $$A$$ is irreducible [7], so in this case prime filters and irreducible filters are the same.…”

Intuitionistic Modal Algebras

“…5.2). Following a similar strategy, we shall use the duality for (implicative) meet semilattices developed in [7,8,13] by Celani and collaborators (Subsect. 5.3) as a basis for our dualities for weak implicative (Subsect.…”

“…Proposition 5.13 below). The following characterizations are quite useful in practice (see [7,8]). A filter F is irreducible iff for every a, b / ∈ F there exists f ∈ F and c / ∈ F such that a ∧ f ≤ c and b ∧ f ≤ c. A filter P is prime iff, for every a, b / ∈ F , there exists c / ∈ F such that a ≤ c and b ≤ c. An order ideal of S is a set I ⊆ S that is decreasing and such that for all a, b ∈ I, there exists c ∈ I with a, b ≤ c. It is easy to see that a filter F is irreducible iff F c = S − F is an order ideal.…”