Stone-Type Representations and Dualities for Varieties of Bisemilattices

Abstract

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn duality and introduce the categories of 2spaces and 2spaces ⋆ . The categories of 2spaces and 2spaces ⋆ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent.Let us recall that, for x ∈ L, (↓x) ⋆ = {F ∈ F : x ′ ∈ F } = ↑x ′ . Since ′ is an involution, it is not difficult to see that ⋆ is a bijection between Y and X. We now show that ⋆ is an order dual mapping.Recall now that (↑x) † = (θ(↑x)) ⋆ .

Condition/Decision Duality and the Internal Logic of Extensive Restriction Categories

“…Like classical logic takes its algebraic semantics in Boolean algebras, the corresponding algebraic structure for K w 3 is that of De Morgan quasilattices (see, e.g., [12]). As is sometimes done, we assume these to be distributive; i.e., what we call De Morgan quasilattices are sometimes called distributive De Morgan quasilattices or even (distributive) De Morgan bisemilattices (see, e.g., [23]). Note that we generally do not require these to be bounded, i.e., for top and bottom elements ⊤ and ⊥ to exist.…”

Logics of left variable inclusion and Płonka sums of matrices

“…The representation theory of regular varieties is largely due to the pioneering work of Płonka [51], and is tightly related to a special class-operator P ł (•) nowadays called Płonka sums. Over the years, regular varieties have been studied in depth both from a purely algebraic perspective [52,39,34,35] and in connection to their topological duals [32,11,60,9,46]. The machinery of Płonka sums has also found useful applications in the study of the constraint satisfaction problem [2] and database semantics [47,56] and in the application of algebraic methods in computer science [13].…”

Dualities for Płonka Sums

“…We recently stated a slightly different duality, still based on P lonka sums, for involutive bisemilattices, see [3] (differences will be briefly explained in Section 3). Dualities for (some) varieties of bisemilattices, although not relying on P lonka sums, are considered in [16].…”