Commutative rings whose ideals form an MV‐algebra

Belluce, L. P.; Nola, Antonio Di

doi:10.1002/malq.200810012

Cited by 29 publications

(27 citation statements)

References 16 publications

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“…(3) As above, this proof can be adapted from that of [1,Proposition 3.12] using the conditions (AN), (CO), (LR). Proof.…”

Section: Generalized Lukasiewicz Ringsmentioning

confidence: 99%

“…The goal of this section is to find a representation of GLRs that would generalize that of Lukasiewicz rings obtained in [1,Theorem 7.7].…”

Section: Representation Of Glrsmentioning

confidence: 99%

“…A ring R is said to be generated by central idempotents, if for every x ∈ R, there exists a central idempotent element e ∈ R such that ex = x. Of these rings, Belluce and Di Nola [1] investigated the commutative rings R for which A(R) is an MV-algebra, which they called Lukasiewicz rings. Recall that MV-algebras, which constitute the algebraic counterpart of Lukasiewicz many value logic are categorically equivalent to Abelian ℓ-groups with strong units [3].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A non-commutative generalization of Łukasiewicz rings

Kadji

Lele

Nganou

2016

Journal of Applied Logic

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The goal of the present article is to extend the study of commutative rings whose ideals form an MV-algebra as carried out by Belluce and Di Nola [1] to non-commutative rings. We study and characterize all rings whose ideals form a pseudo MV-algebra, which shall be called here generalized Lukasiewicz rings. We obtain that these are (up to isomorphism) exactly the direct sums of unitary special primary rings.

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“…(3) As above, this proof can be adapted from that of [1,Proposition 3.12] using the conditions (AN), (CO), (LR). Proof.…”

Section: Generalized Lukasiewicz Ringsmentioning

confidence: 99%

“…The goal of this section is to find a representation of GLRs that would generalize that of Lukasiewicz rings obtained in [1,Theorem 7.7].…”

Section: Representation Of Glrsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A non-commutative generalization of Łukasiewicz rings

Kadji

Lele

Nganou

2016

Journal of Applied Logic

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“…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

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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.

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“…This was started by Di Nola and Gerla [10] who defined an MV-semiring attached to an MV-algebra. Belluce and Di Nola [4] simplified it to an equivalent definition of MV-semirings. These two authors and Ferraioli [5] then established a categorical equivalence between MV-algebras and MV-semirings and used it to obtain representation of MV-algebras as certain spaces of continuous functions via a corresponding representation of MV-semirings.…”

Section: Introductionmentioning

confidence: 99%

Lattice-ordered abelian groups finitely generated as semirings

Kala¹

2017

J. Commut. Algebra

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Abstract. A lattice-ordered group (an ℓ-group) G(⊕, ∨, ∧) can be naturally viewed as a semiring G(∨, ⊕). We give a full classification of (abelian) ℓ-groups which are finitely generated as semirings, by first showing that each such ℓ-group has an order-unit so that we can use the results of Busaniche, Cabrer and Mundici [8]. Then we carefully analyze their construction in our setting to obtain the classification in terms of certain ℓ-groups associated to rooted trees (Theorem 4.1).This classification result has a number of important applications: for example it implies a classification of finitely generated ideal-simple (commutative) semirings S(+, ·) with idempotent addition and provides important information concerning the structure of general finitely generated ideal-simple (commutative) semirings, useful in obtaining further progress towards Conjecture 1.1 discussed in [2], [15].

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Commutative rings whose ideals form an MV‐algebra

Cited by 29 publications

References 16 publications

A non-commutative generalization of Łukasiewicz rings

A non-commutative generalization of Łukasiewicz rings

Idempotence of finitely generated commutative semifields

Lattice-ordered abelian groups finitely generated as semirings

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