Global determinism of completely regular semigroups

TL;DR

The paper investigates whether completely regular semigroups are globally determined by their power semigroups $P(S)$, i.e., whether $P(S)\cong P(S')$ implies $S\cong S'$. It develops a componentwise analysis via subsets $\mathcal{A}_2(S)$ and $\mathcal{A}_3(S)$, examines the structure semilattice $Y=S/\mathscr{D}$ with a semilattice isomorphism $\theta:Y\to Y'$, and analyzes images of singleton sets to construct an isomorphism $\eta:S\to S'$. The main result proves that the class $\mathcal{CR}$ of completely regular semigroups is globally determined, thereby extending several earlier global-determinism results. This advances understanding of how power-semigroup invariants capture the full algebraic structure of completely regular semigroups and their Green’s-structure decomposition.

Abstract

The power semigroup of a semigroup $ S $ is the semigroup of all nonempty subsets of $ S $ equipped with the naturally defined multiplication. A class $\mathcal{K} $ of semigroups is globally determined if any two members of $ \mathcal{K} $ with isomorphic globals are themselves isomorphic. The global determinability for various classes of semigroups has attracted some attention during the past 50 years. In this paper we prove that the class of all completely regular semigroups is globally determined. This is an extension and generalization of a series of related results obtained by some other mathematicians.

Theorems & Definitions (45)

• Proposition 1.1
• Lemma 1.2: redei
• Lemma 1.3: pelikan
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• Theorem 2.5