Orthogonal relational systems

Bonzio, Stefano; Chajda, Ivan; Ledda, Antonio

doi:10.1007/s00500-015-1999-4

Cited by 2 publications

(3 citation statements)

References 17 publications

(24 reference statements)

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“…,0,1⟩ is an orthomodular near semiring. 4 We can also prove the converse, stating a correspondence between orthomodular lattices and orthomodular near semirings.…”

Section: Lemmamentioning

confidence: 91%

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Representing quantum structures as near semirings

2016

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In this article, we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called Łukasiewicz near semirings, as a particular case of near semirings, and we show that every basic algebra is representable as (precisely, it is term equivalent to) a near semiring. In the particular case in which a Łukasiewicz near semiring is also a semiring, we obtain as a corollary a representation of MV-algebras as semirings. Analogously, by introducing a particular subclass of Łukasiewicz near semirings, that we termed orthomodular near semirings, we obtain a representation of orthomodular lattices. In the second part of the article, we discuss several universal algebraic properties of Łukasiewicz near semirings and we show that the variety of involutive integral near semirings is a Church variety. This yields a neat equational characterization of central elements of this variety. As a byproduct of such, we obtain several direct decomposition theorems for this class of algebras.

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“…,0,1⟩ is an orthomodular near semiring. 4 We can also prove the converse, stating a correspondence between orthomodular lattices and orthomodular near semirings.…”

Section: Lemmamentioning

confidence: 91%

“…Diverse applications of this theory can be found in[4,14].at UniversitÃ di Cagliari on June 13, 2016 http://jigpal.oxfordjournals.org/ Downloaded from…”

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confidence: 99%

Representing quantum structures as near semirings

2016

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“…The study of binary relations traces back to the work of J. Riguet [14], while a first attempt to provide an algebraic theory of relational systems is due to Mal'cev [12]. Relational systems of different kinds have been investigated by different authors for a long time, see for example [4], [3], [8], [9], [10]. Binary relational systems are very important for the whole of mathematics, as relations, and thus relational systems, represent a very general framework appropriate for the description of several problems, which can turn out to be useful both in mathematics and in its applications.…”

Section: Introductionmentioning

confidence: 99%

Residuated relational systems

Bonzio

Chajda

2018

Asian-European J. Math.

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The aim of the present paper is to generalize the concept of residuated poset, by replacing the usual partial ordering by a generic binary relation, giving rise to relational systems which are residuated. In particular, we modify the definition of adjointness in such a way that the ordering relation can be harmlessly replaced by a binary relation. By enriching such binary relation with additional properties we get interesting properties of residuated relational systems which are analogical to those of residuated posets and lattices. X * = {y ∈ A : (x, y) ∈ R, for each x ∈ X}, X † = {x ∈ A : (x, y) ∈ R, for each y ∈ X}.

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Orthogonal relational systems

Cited by 2 publications

References 17 publications

Representing quantum structures as near semirings

Representing quantum structures as near semirings

Residuated relational systems

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