Representing quantum structures as near semirings

Bonzio, Stefano; Chajda, Ivan; Ledda, Antonio

doi:10.1093/jigpal/jzw031

Cited by 9 publications

(15 citation statements)

References 14 publications

(34 reference statements)

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“…), with e a central element. This theorem subsumes analogous results for basic algebras, orthomodular lattices, and MV-algebras, since these structures are term-equivalent subvarieties of the variety of involutive idempotent integral near semirings (see [2]).…”

Section: Introductionsupporting

confidence: 62%

“…Our next task will be to provide a full description of the principal ideals of Id(A), for any Lukasiewicz near semiring A. As it was proved in [2], the variety of Lukasiewicz near semirings is congruence-permutable, as witnessed by the Mal'cev term…”

Section: Ideals In Lukasiewicz Near Semiringsmentioning

confidence: 99%

“…Let us remark that in general, cf [2]. and[4], the notion of involutive semiring do not involve integrality.Indeed, there are involutive semirings which are not integral (cf [2,. Remark 3]).…”

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confidence: 99%

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On the structure theory of Łukasiewicz near semirings

Chajda

Fazio

Ledda

2017

Logic Journal of the IGPL

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In a previous article by two of the present authors and S. Bonzio, Lukasiewicz near semirings were introduced and it was proven that basic algebras can be represented (precisely, are term equivalent to) as near semirings. In the same work it has been shown that the variety of Lukasiewicz near semirings is congruence regular. In other words, every congruence is uniquely determined by its 0-coset. Thus, it seems natural to wonder wether it could be possible to provide a set-theoretical characterization of these cosets. This article addresses this question and shows that kernels can be neatly described in terms of two simple conditions. As an application, we obtain a concise characterization of ideals in Lukasiewicz semirings. Finally, we close this article with a rather general Cantor-Bernstein type theorem for the variety of involutive idempotent integral near semirings.

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Section: Introductionsupporting

confidence: 62%

Section: Ideals In Lukasiewicz Near Semiringsmentioning

confidence: 99%

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On the structure theory of Łukasiewicz near semirings

Chajda

Fazio

Ledda

2017

Logic Journal of the IGPL

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“…As we mentioned in the introduction, bisemilattices were firstly considered by J. P lonka as a common generalization of semilattices and lattices, and, over the years, they have kindled the attention of several scholars. An extensive guide to the bibliography on semirings -of which distributive bisemilattices form a prominent subvariety -can be found in K. Glazek's book [20] (for recent developments the interested reader may also consult [7,24,25]).…”

Section: Bisemilatticesmentioning

confidence: 99%

Stone-Type Representations and Dualities for Varieties of Bisemilattices

Ledda

2017

Stud Logica

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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)) ⋆ .

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“…These semirings originated in the study of quantum structures, see, for example, Bonzio et al. ( 2016 ) and Chajda et al. ( 2018 ) for the concepts and motivation.…”

Section: Introductionmentioning

confidence: 99%

Ideals and their complements in commutative semirings

Chajda

Länger

2018

Soft Comput

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We study conditions under which the lattice of ideals of a given a commutative semiring is complemented. At first we check when the annihilator of a given ideal I of is a complement of I . Further, we study complements of annihilator ideals. Next we investigate so-called Łukasiewicz semirings. These form a counterpart to MV-algebras which are used in quantum structures as they form an algebraic semantic of many-valued logics as well as of the logic of quantum mechanics. We describe ideals and congruence kernels of these semirings with involution. Finally, using finite unitary Boolean rings, a construction of commutative semirings with complemented lattice of ideals is presented.

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Representing quantum structures as near semirings

Cited by 9 publications

References 14 publications

On the structure theory of Łukasiewicz near semirings

On the structure theory of Łukasiewicz near semirings

Stone-Type Representations and Dualities for Varieties of Bisemilattices

Ideals and their complements in commutative semirings

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