Some Study of Semigroups of h -Bi-Ideals of Semirings

Semigroups are generalizations of groups and rings. In the semigroup theory, there are certain kinds of band decompositions which are useful in the study of the structure of semigroups. This research will open up new horizons in the field of mathematics by aiming to use semigroup of h -bi-ideal of semiring with semilattice additive reduct. With the course of this research, it will prove that subsemigroup, the set of all right h -bi-ideals, and set of all left h -bi-ideals are bands for h -regular semiring. Moreover, it will be demonstrated that if semigroup of all h -bi-ideals B H , ∗ is semilattice, then H is h -Clifford. This research will also explore the classification of minimal h -bi-ideal.

Additively orthodox semirings with special transversals

<abstract><p>A semiring $ (S, +, \cdot) $ is called additively orthodox semiring if its additive reduct $ (S, +) $ is a orthodox semigroup. In this paper, by introducing some special semiring transversals as the tools, the constructions of additively orthodox semirings with a skew-ring transversal or with a generalized Clifford semiring transversal are established. Meanwhile, it is shown that an additively orthodox semiring with a generalized Clifford semiring transversal is a b-lattice of additively orthodox semirings with skew-ring transversals. Consequently, the corresponding results of Clifford semirings and generalized Clifford semirings in reference (M. K. Sen, S. K. Maity, K. P. Shum, Clifford semirings and generalized Clifford semirings, Taiwan. J. Math., 9 (2005), 433–444) and completely regular semirings in reference (S. K. Maity, M. K. Sen, K. P. Shum, On completely regular semirings, Bull. Cal. Math. Soc., 98 (2006), 319–328) are extended and strengthened.</p></abstract>

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

We 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

THE ZELEZNIKOW PROBLEM ON A CLASS OF ADDITIVELY IDEMPOTENT SEMIRINGS

A 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.

The Structure of Almost Idempotent Semirings

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].