Duality for distributive bisemilattices

Gierz, Gerhard; Romanowska, Anna

doi:10.1017/s1446788700034224

Cited by 16 publications

(13 citation statements)

References 19 publications

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“…For the second part, by [4], we have that S is a GR space, thus we only have to check that ¬ has the required properties. Let ϕ, ψ ∈ S and x ∈ S; properties G1 − G4 can be easily verified as follows:…”

Section: The Category Of Involutive Bisemilattices and Its Dualmentioning

confidence: 99%

A Duality for Involutive Bisemilattices

2018

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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.

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Section: The Category Of Involutive Bisemilattices and Its Dualmentioning

confidence: 99%

A Duality for Involutive Bisemilattices

2018

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“…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.…”

Section: The Dualitymentioning

confidence: 99%

Dualities for Płonka Sums

Bonzio

2018

Log. Univers.

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P lonka sums consist of an algebraic construction similar, in some sense to direct limits, which allows to represent classes of algebras defined by means of regular identities (namely those equations where the same set of variables appears on both sides). Recently, P lonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a P lonka sum representation in terms of dualisable algebras.2010 Mathematics Subject Classification. Primary 08C20. Secondary 06E15, 18A99, 22A30.

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“…Note that the 'natural dualities' considered in [3,6,8,9,18] are of a similar type. However, they require finite schizophrenic objects and satisfaction of certain additional conditions that are not necessarily obtained for the cases considered in this paper.…”

Section: Dualitymentioning

confidence: 99%

Duality for Quasipolytopes

Mućka¹,

Romanowska²

2016

J. Aust. Math. Soc.

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In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. The present paper provides an extension of this duality to a much more general class of so-called quasipolytopes, that is, convex sets with polytopes as closures. The dual spaces of quasipolytopes are Płonka sums of open polytopes, which are considered as barycentric algebras with some additional operations. In constructing this duality, we use several known and new dualities: the Hofmann–Mislove–Stralka duality for semilattices; the Romanowska–Ślusarski–Smith duality for polytopes; a new duality for open polytopes; and a new duality for injective Płonka sums of polytopes.

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Duality for distributive bisemilattices

Abstract: We establish a duality between distributive bisemilattices and certain compact left normal bands. The main technique in the proof utilizes the idea of Plonka sums.

Cited by 16 publications

References 19 publications

A Duality for Involutive Bisemilattices

A Duality for Involutive Bisemilattices

Dualities for Płonka Sums

Duality for Quasipolytopes

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