[Paper Review] On completely regular and strongly regular ordered $\Gamma$-semigroups

Niovi Kehayopulu|arXiv (Cornell University)|Jul 16, 2013

Fuzzy and Soft Set Theory1 references3 citations

TL;DR

This paper establishes characterizations of completely regular and strongly regular ordered Γ-semigroups by extending concepts from ordered semigroups. It introduces strongly regular po-Γ-semigroups, proves that complete regularity is equivalent to bi-ideals being semiprime, and shows that strong regularity implies left, right, and two-sided regularity, with a key equivalence linking strong regularity to the structure of subsemigroups (MΓaΓM).

ABSTRACT

Our aim is to show the way we pass from the results of ordered semigroups (or semigroups) to ordered $\\Gamma$-semigroups (or $\\Gamma$-semigroups). The results of this note have been transferred from ordered semigroups. The concept of strongly regular $po$-$\\Gamma$-semigroups has been first introduced here and a characterization of strongly regular $po$-$\\Gamma$-semigroups is given.

Motivation & Objective

• To extend results from ordered semigroups to ordered Γ-semigroups by adapting definitions and properties.
• To define and investigate the concept of strongly regular po-Γ-semigroups, a novel class introduced in this work.
• To establish equivalent characterizations of complete and strong regularity in po-Γ-semigroups using bi-ideals and subsemigroup structures.
• To demonstrate that strong regularity implies left, right, and two-sided regularity in po-Γ-semigroups.
• To show that complete regularity is equivalent to every bi-ideal being semiprime, and to relate this to the equality of generated bi-ideals.

Proposed method

• Adapts definitions of bi-ideals in po-Γ-semigroups: a subset B is a bi-ideal if BΓMΓB ⊆ B and closed under order (if b ≤ a and b ∈ M, then b ∈ B).
• Uses the concept of generated bi-ideals: B(A) = (A ∪ AΓAΓA] for a subset A, with B(a) = (a ∪ aΓaΓa].
• Applies the key condition: a po-Γ-semigroup M is completely regular iff for every a ∈ M, there exist x ∈ M and γ,μ,ρ,ξ ∈ Γ such that a ≤ (aγa)μxρ(aξa).
• Establishes equivalence between complete regularity and bi-ideals being semiprime: if aΓa ⊆ B then a ∈ B.
• Proves that complete regularity is equivalent to B(a) = B(aΓa) = B(aΓaΓMΓaΓa) for all a ∈ M.
• Introduces strong regularity via: a ≤ aγxμa and aγx = xγa = xμa = aμx for some x ∈ M and γ,μ ∈ Γ, and proves its implications.

Experimental results

Research questions

• RQ1 What is the appropriate generalization of strong regularity to po-Γ-semigroups, and how does it relate to regularity?
• RQ2 How can the concept of bi-ideals in po-Γ-semigroups be used to characterize complete regularity?
• RQ3 What is the relationship between the generated bi-ideals B(a), B(aΓa), and B(aΓaΓMΓaΓa) in a completely regular po-Γ-semigroup?
• RQ4 Under what conditions is a po-Γ-semigroup strongly regular, and how does this relate to the regularity of subsemigroups (MΓaΓM)?
• RQ5 Is the condition B = (BΓB] for all bi-ideals B sufficient to imply regularity in a po-Γ-semigroup?

Key findings

• A po-Γ-semigroup M is completely regular if and only if for every a ∈ M, there exist x ∈ M and γ,μ,ρ,ξ ∈ Γ such that a ≤ (aγa)μxρ(aξa).
• Complete regularity in a po-Γ-semigroup is equivalent to every bi-ideal being semiprime: if aΓa ⊆ B, then a ∈ B.
• For a completely regular po-Γ-semigroup, the generated bi-ideals satisfy B(a) = B(aΓa) = B(aΓaΓMΓaΓa) for all a ∈ M.
• In a completely regular po-Γ-semigroup, every bi-ideal B satisfies B = (BΓB], and conversely, if B = (BΓB] for all bi-ideals, then M is regular.
• A po-Γ-semigroup M is strongly regular if and only if it is left regular, right regular, and (MΓaΓM] is strongly regular for every a ∈ M.
• Strong regularity implies that for every a ∈ M, there exists x ∈ M and γ,μ ∈ Γ such that a ≤ aγxμa and aγx = xγa = xμa = aμx.