MV-semirings and their Sheaf Representations

Belluce, L. P.; Nola, Antonio Di; Ferraioli, Anna Rita

doi:10.1007/s11083-011-9234-0

Cited by 26 publications

(28 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Yet another class of applications arises thanks to the correspondence between certain semifields, latticeordered groups, and MV-algebras. These provide useful tools in multi-valued logic [1,2,3,9,10,30,31]. For further applications and references, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Idempotence of finitely generated commutative semifields

Kala

Korbelář

2018

Forum Mathematicum

View full text Add to dashboard Cite

We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.2010 Mathematics Subject Classification. primary: 12K10, 16Y60, 20M14, secondary: 11H06, 52A20.This statement can be now understood as an extension of the folklore theorem referred in the beginning, i.e., that every (commutative) field that is finitely generated as a ring is finite. Of course, the greatest difference is the existence of additively idempotent semifields.Since every semifield is clearly ideal-simple, part (b) of the theorem follows immediately from (c). Also, one can be slightly more precise in the additively constant case: this occurs if and only if there is a finitely generated (multiplicative) abelian group G(·) and the semiring is the semifield S := G ∪ {o}, where o is a new element. Operations that extend the multiplication on G are defined by a+b = o and a·o = o for all a, b ∈ S.

show abstract

Section: Introductionmentioning

confidence: 99%

Idempotence of finitely generated commutative semifields

Kala

Korbelář

2018

Forum Mathematicum

View full text Add to dashboard Cite

show abstract

“…A representation of MV-algebras by means of semiring-like structures, so-called MV-semirings, was already published by Belluce et al in [1] and it was further developed by Di Nola and Russo in [6]. These facts encouraged us to try a similar approach also for lattice effect algebras.…”

mentioning

confidence: 99%

“…These facts encouraged us to try a similar approach also for lattice effect algebras. The motivation for such a representation by means of semiring-like structures is practically the same as in [1]. The fact that the binary operation in effect algebras is only partial and that one cannot suppose that, whenever extended to a total operation, it would remain commutative and associative, motivated us to use so-called right near semirings instead of semirings.…”

mentioning

confidence: 99%

See 1 more Smart Citation

A Representation of Lattice Effect Algebras by Means of Right Near Semirings with Involution

Chajda

Länger

2016

Int J Theor Phys

View full text Add to dashboard Cite

Since every lattice effect algebra decomposes into blocks which are MV-algebras and since every MV-algebra can be represented by a certain semiring with an antitone involution as shown by Belluce, Di Nola and Ferraioli, the natural question arises if a lattice effect algebra can also be represented by means of a semiring-like structure. This question is answered in the present paper by establishing a one-to-one correspondence between lattice effect algebras and certain right near semirings with an antitone involution.Keywords Effect algebra · Lattice effect algebra · Right near semiring · Antitone involution · Effect near semiring Effect algebras were introduced by Foulis and Bennett [8] in order to axiomatize unsharp logics of quantum mechanics. Although the definition of an effect algebra looks elementary, these algebras have several very surprising properties. Concerning these properties the reader is referred to the monograph [7] by Dvurečenskij and Pulmannová. In particular, every effect algebra induces a natural partial order relation and thus can be considered as a bounded poset. If this poset is a lattice, the effect algebra is called a lattice effect algebra.

show abstract

“…(ii) (L, ⊗, 1) is a commutative monoid. 1 Support of the research of both authors by the bilateral project "New perspectives on residuated posets", supported by the Austrian Science Fund (FWF), project I 1923-N25, and the Czech Science Foundation (GAČR), project 15-34697L, as well as by the project "Ordered structures for algebraic logic", supported by AKTION Austria -Czech Republic, project 71p3, is gratefully acknowledged. As a source for elementary properties of residuated lattices see the monograph by Bělohlávek ([2]).…”

mentioning

confidence: 99%