Idempotent semirings with a commutative additive reduct

Xianzhong Zhao

doi:10.1007/s002330010048

THE ZELEZNIKOW PROBLEM ON A CLASS OF ADDITIVELY IDEMPOTENT SEMIRINGS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788713000359 ◽

2013 ◽

Vol 95 (3) ◽

pp. 404-420

Author(s):

YONG SHAO ◽

SINIŠA CRVENKOVIĆ ◽

MELANIJA MITROVIĆ

Keyword(s):

Semigroup Forum ◽

Inverse Semigroup ◽

Regular Ring ◽

Clifford Semigroup ◽

Binary Operations ◽

Idempotent Semirings ◽

Additive Reduct

AbstractA semiring is a set $S$ with two binary operations $+ $ and $\cdot $ such that both the additive reduct ${S}_{+ } $ and the multiplicative reduct ${S}_{\bullet } $ are semigroups which satisfy the distributive laws. If $R$ is a ring, then, following Chaptal [‘Anneaux dont le demi-groupe multiplicatif est inverse’, C. R. Acad. Sci. Paris Ser. A–B 262 (1966), 274–277], ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. In Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136], it is proved that if $R$ is a regular ring then ${R}_{\bullet } $ is orthodox if and only if ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. The latter result, also known as Zeleznikow’s theorem, does not hold in general even for semirings $S$ with ${S}_{+ } $ a semilattice Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136]. The Zeleznikow problem on a certain class of semirings involves finding condition(s) such that Zeleznikow’s theorem holds on that class. The main objective of this paper is to solve the Zeleznikow problem for those semirings $S$ for which ${S}_{+ } $ is a semilattice.

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The Structure of Almost Idempotent Semirings

Algebra Colloquium ◽

10.1142/s1005386710000799 ◽

2010 ◽

Vol 17 (spec01) ◽

pp. 851-864 ◽

Cited By ~ 2

Author(s):

M. K. Sen ◽

A. K. Bhuniya

Keyword(s):

Distributive Lattice ◽

Idempotent Semiring ◽

Idempotent Semirings ◽

Additive Reduct

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 [Formula: see text] of all almost idempotent semirings corresponding to the eight subvarieties of the variety [Formula: see text] 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 [Formula: see text]. This semiring Pf(F) is free in a subclass of [Formula: see text] if and only if F is in the corresponding subvariety of [Formula: see text].

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Exploring positively invariant sets by linear systems over idempotent semirings

IMA Journal of Mathematical Control and Information ◽

10.1093/imamci/21.3.307 ◽

2004 ◽

Vol 21 (3) ◽

pp. 307-322 ◽

Cited By ~ 3

Author(s):

L. Truffet

Keyword(s):

Linear Systems ◽

Invariant Sets ◽

Idempotent Semirings ◽

Positively Invariant Sets

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Semigroup of k-bi-Ideals of a Semiring with Semilattice Additive Reduct

Demonstratio Mathematica ◽

10.1515/dema-2016-0011 ◽

2016 ◽

Vol 49 (2) ◽

Author(s):

A. K. Bhuniya ◽

K. Jana

Keyword(s):

Regular Semigroup ◽

Normal Band ◽

Additive Reduct

AbstractWe associate a semigroup B(S) to every semiring S with semilattice additive reduct, namely the semigroup of all k-bi-ideals of S; and such semirings S have been characterized by this associated semigroup B(S). A semiring S is k-regular if and only if B(S) is a regular semigroup. For the left k-Clifford semirings S, B(S) is a left normal band; and consequently, B(S) is a semilattice if S is a k-Clifford semiring. Also we show that the set B

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