Green's relations in Partial \(\Gamma\)-Semirings

Abstract

A \(\Gamma\)-so-ring is a structure possessing a natural partial ordering, an infinitary partial addition and a ternary multiplication, subject to a set of axioms. The partial functions under disjoint-domain sums and functional composition is a \(\Gamma\)-so-ring. In this paper we introduce the notions of dense and annihilator ideals of \(\Gamma\)-so-rings and obtain the Green's relations in Partial \(\Gamma\) semirings.

REGULAR $\Gamma$-SO-RINGS

“…In this section we collect important definitions and results from [7], [12], [14], [15], [16], [17] and [18]. Definition 2.1.…”

“…Definition 2.6. [12] A partial Γ-semiring R is said be a sum-ordered partial Γ-semiring (in short Γ-so-ring) if the partial monoids R and Γ are somonoids.…”

“…[15] Let R be a Γ-so-ring with left/right unity and a, b ∈ R. Then a ∈ aΓR and b ∈ RΓb. An ideal A of a Γ-so-ring R is called proper if A = R.…”

“…Murali Krishna Rao [11] in 1995 introduced the notion of a Γ-semiring as a generalization of semirings and Γ-rings, and extended many fundamental results of semirings and Γ-rings to Γ-semirings. In [12] and [13] we introduced the notion of Γ-so-ring, obtained a necessary and sufficient condition for the quotient R/θ to be a Γ/σ-so-ring, where (θ, σ) is a congruence relation on (R, Γ) and (φ, ρ)-representation of Γ-so-rings. In [14], [15], [16] and [17], we intoduced the notions of an ideal, a prime ideal and a semiprime ideal in a Γ-so-ring R and obtained many characteristics of them in R. As a continuation, in this paper we introduce the notions of irreducible and strongly irreducible ideals and a regular Γ-so-ring and we obtain the characteristics of them.…”