A Duality for Involutive Bisemilattices

Abstract

We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as P lonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be axiomatized as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska, equipped with an involution as additional operation.

A Simple Logical Matrix and Sequent Calculus for Parry’s Logic of Analytic Implication

“…In the second, we show that some features of said semantics can be generalized in order to provide matrices for similar fragments of other logics-fragments that also respect Parry's Proscriptive Principle. 7 3.1. P lonka sums P lonka sums are operations on algebras that allow to define new structures from a previously given collection thereof.…”

Dualities for Płonka Sums

“…A related question concerns the possibility of describing semilattice inverse systems of topological spaces as a unique space. This is done in some known special cases, as distributive bisemilattices [13], the P lonka sum of distributive lattices and involutive bisemilattices [3], the P lonka sum of Boolean algebras.…”

“…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].…”

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].…”