Abstract
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.
Highlights
Primary idea of semigroup and monoid is given by Wallis [1]
Semirings play a significant role in geometry, but they play a role in pure mathematics
We will work on construction of semigroups of h-bi-ideals in h-regular semiring in Section 3, and Section 4 will consist of semigroup of h-bi-ideals in h-Clifford semiring, a discussion on semigroup of minimal h-bi-ideal
Summary
Primary idea of semigroup and monoid is given by Wallis [1]. Wallis expressed that a set which satisfies the associative law under some binary operations is called semigroup, and a set which is semigroup with identity is called monoid. The concept of regular semirings was introduced by Von Neumann [23] and Bourne [24]. Bourne showed that if ∀t ∈ H, there exists h1, h2 ∈ H such that t + th1t = th2t, the semiring is regular It was in 1952 that Good and Hughes [25] first described the definition of a bi-ideal of semigroup. We have shown that the h -bi-ideal semigroup BðHÞ elegantly illustrates different sectors of a semiring. We will define the different subclass of the h-regular semiring by their semigroup of h-bi-ideals. We will characterize a new class of semigroup of h-bi-ideals in h-Clifford semiring. We will work on construction of semigroups of h-bi-ideals in h-regular semiring, and Section 4 will consist of semigroup of h-bi-ideals in h-Clifford semiring, a discussion on semigroup of minimal h-bi-ideal.
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