A semiring-like representation of lattice pseudoeffect algebras

Chajda, Ivan; Fazio, Davide; Ledda, Antonio

doi:10.1007/s00500-018-3157-2

Cited by 2 publications

(1 citation statement)

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“…c 2018 Miskolc University Press unary operation. Moreover, Łukasiewicz semirings are additively idempotent and ordered by the induced semilattice ordering, see [1] - [4]. On the other hand, these semirings are not lattices and hence we cannot use results and methods from [7] or [11].…”

Section: Introductionmentioning

confidence: 99%

Derivations in Lukasiewicz semirings

Chajda¹,

Länger²

2018

MMN

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An axiomatization of classical propositional logic is provided by means of Boolean algebras which are term equivalent to Boolean rings. This is important because rings form a classical part of algebra whose tools can be used for the investigations. The Łukasiewicz manyvalued logic was axiomatized via so-called MV-algebras by C. C. Chang in 1950's. MV-algebras are successfully applied in the logic of quantum mechanics and hence they are considered as quantum structures. It is a natural question if also MV-algebras have their alter ego among classical structures. For this reason the so-called Łukasiewicz semirings were introduced by the first author and his collaborators in [3]-[4]. As shown, Łukasiewicz semirings are term equivalent to MV-algebras and we can use with advantage several developed tools for their study. In particular, we investigate derivations in semirings which were introduced formerly but here these semirings are enriched by an involution.

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Section: Introductionmentioning

confidence: 99%

Derivations in Lukasiewicz semirings

Chajda¹,

Länger²

2018

MMN

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On EMV-Semirings

Borzooei

Shenavaei

Nola

et al. 2019

Mathematica Slovaca

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The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI-simple and DLO-semiring and prove that every EMV-semiring is a GI-simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.

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A semiring-like representation of lattice pseudoeffect algebras

Cited by 2 publications

References 18 publications

Derivations in Lukasiewicz semirings

Derivations in Lukasiewicz semirings

On EMV-Semirings

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