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Journal ArticleDOI

The structure of almost idempotent semirings

01 Dec 2010-Algebra Colloquium (Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University)-Vol. 17, pp 851-864
TL;DR: In this article, the authors introduced the notion of almost idempotent semirings, which are semiring with semilattice additive reduct satisfying the identity x + x2 = x2.
Abstract: In this paper we introduce the notion of almost idempotent semirings as the semirings with semilattice additive reduct satisfying the identity x + x2 = x2, and characterize eight subclasses of the variety of all almost idempotent semirings corresponding to the eight subvarieties of the variety of all normal bands. Every almost idempotent semiring S is a distributive lattice of rectangular almost idempotent semirings. Given a semigroup F, the semiring Pf(F) of all finite non-empty subsets of F is almost idempotent precisely when F is a band, and in this case, Pf(F) is freely generated by the band F in the variety . This semiring Pf(F) is free in a subclass of if and only if F is in the corresponding subvariety of .
Citations
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13 May 2013

16 citations


Cites background from "The structure of almost idempotent ..."

  • ...Sen and Bhuniya introduced in [SB10] the notion of almost idempotent semirings....

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Journal ArticleDOI
TL;DR: It is well known that if δ : S → S is a derivation in semiring S, then in the semiring Mn(S) of n × n matrices over S, the map δher(A) = (δ(aij)) for any matrix A = (aij) ∈ Mn (S) is...
Abstract: It is well known that if δ : S → S is a derivation in semiring S, then in the semiring Mn(S) of n × n matrices over S, the map δher such that δher(A) = (δ(aij)) for any matrix A = (aij) ∈ Mn(S) is ...

5 citations

Journal ArticleDOI
TL;DR: In this article, the authors characterize the subsemiring of a k-regular semiring S with commutative addition with a semilattice additive reduct as an almost idempotent.
Abstract: An element e of a semiring S with commutative addition is called an almost idempotent if $$e + e^2 = e^2$$ . Here we characterize the subsemiring $$\langle E(S)\rangle $$ generated by the set E(S) of all almost idempotents of a k-regular semiring S with a semilattice additive reduct. If S is a k-regular semiring then $$\langle E(S)\rangle $$ is also k-regular. A similar result holds for the completely k-regular semirings, too.

3 citations

Journal ArticleDOI
TL;DR: In this article, left k-Clifford semirings were introduced as a generalization of k-clifford semiirings and a semiring is a left-k-semifield semiring if and only if it is a distributive lattice of left-kliffords.
Abstract: Here, being motivated from the works of Zhu, Guo and Shum on left Clifford semigroups, we have introduced left k-Clifford semirings as a generalization of k-Clifford semirings. A k-regular semiring S ∈ 𝕊𝕃+ is a left k-Clifford semiring (left k-semifield) if for all a ∈ S, (for all b ∈ S′, ). Several characteristics for this class of semirings are obtained. Moreover a semiring is left k-Clifford if and only if it is a distributive lattice of left k-semifields.

2 citations

01 Jan 2013

1 citations

References
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Book
01 Jan 1995
TL;DR: Inverse semigroups as discussed by the authors are a subclass of regular semigroup classes and can be seen as semigroup amalgamations of semigroup groups, which is a special case of regular semiigroups.
Abstract: 1. Introductory ideas 2. Green's equivalences regular semigroups 3. 0-simple semigroups 4. Completely regular semigroups 5. Inverse semigroups 6. Other classes of regular semigroups 7. Free semigroups 8. Semigroup amalgams References List of symbols

1,979 citations

Book ChapterDOI
27 Feb 1997
TL;DR: Exotic semirings have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis.
Abstract: Exotic semirings such as the “(max, +) semiring” (ℝ ∪ {−∞},max,+), or the “tropical semiring” (ℕ ∪ {+∞}, min, +), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities).

243 citations

Journal ArticleDOI
TL;DR: In this paper, a systematic equational classification of semilattice-ordered semigroups is presented, and the classification is reduced to finding certain operators on relatively free semiigroups.
Abstract: We initiate here a systematic equational classification of semilattice-ordered semigroups. We start with various examples. Then we consider semilattice-ordered semigroups satisfying the identity xr = x. Finally, we reduce the classification to finding certain operators on relatively free semigroups.

31 citations