On injectivity of semimodules over additively idempotent division semirings and chain MV-semirings

Nola, Antonio Di; Lenzi, Giacomo; Nam, Tran Giang; Vannucci, Sara

doi:10.1016/j.jalgebra.2019.07.025

Cited by 3 publications

(2 citation statements)

References 29 publications

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“…generalizing similar result for MV-semirings in [2]. Indeed, involution seems to play a crucial role and the results for MV-semirings seem to be generalizable whenever the Mundici functor is not involved.…”

Section: Introductionsupporting

confidence: 75%

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Injective and projective semimodules over involutive semirings

Jipsen¹,

Vannucci²

2020

Preprint

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We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective helps us find a necessary and sufficient condition for the interval [0, 1] to be a subalgebra of an involutive residuated lattice. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.

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Section: Introductionsupporting

confidence: 75%

“…Proposition 4.11 in [3] shows that MV-algebras and MV-semirings are termequivalent. This result has led to fruitful interaction between research in fuzzy logic and semirings/semimodules [2,3]. Below we show that the term-equivalence lifts to involutive residuated lattices and a class of non-commutative 0-free semirings.…”

Section: Involutive Semiringsmentioning

confidence: 65%

Injective and projective semimodules over involutive semirings

Jipsen¹,

Vannucci²

2020

Preprint

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show abstract

Injective and projective semimodules over involutive semirings

Jipsen¹,

Vannucci²

2021

J. Algebra Appl.

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We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval [Formula: see text] to be a subalgebra of an involutive residuated lattice, where [Formula: see text] is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.

show abstract

On Fuzzy Proper Exact Sequences and Fuzzy Projective Semimodules Over Semirings

Sahni¹,

Pandey²,

Mishra

et al. 2021

Wseas Transactions on Mathematics

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As an analogue here we extend and give new horizon to semimodule theory by introducing fuzzy exact and proper exact sequences of fuzzy semi modules for generalizing well known theorems and results of semimodule theory to their fuzzy environment. We also elucidate completely the characterization of fuzzy projective semi modules via Hom functor and show that semimodule µP is fuzzy projective if and only if Hom(µP ,–) preservers the exactness of the sequence µM′ α¯−→νM β¯ −→ηM′′ with β¯ being K-regular. Some results of commutative diagram of R-semimodules having exact rows specifically the “5-lemma” to name one, were easily transferable with the novel proofs in their fuzzy context. Also, towards the end apart from the other equivalent conditions on homomorphism of fuzzy semimodules it is necessary to see that in semimodule theory every fuzzy free is fuzzy projective however the converse is true only with a specific condition.

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On injectivity of semimodules over additively idempotent division semirings and chain MV-semirings

Cited by 3 publications

References 29 publications

Injective and projective semimodules over involutive semirings

Injective and projective semimodules over involutive semirings

Injective and projective semimodules over involutive semirings

On Fuzzy Proper Exact Sequences and Fuzzy Projective Semimodules Over Semirings

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