Epimorphisms and pseudovarieties

TL;DR

The paper investigates when epimorphisms between semigroups preserve membership in pseudovarieties, introducing dominion, saturation, and F-saturation as central notions. It develops general tools (Isbell’s Zigzag, stability) and proves that DS-structured semigroups have only onto epimorphisms, while the families $\mathsf{V}_i$ (for $i=1,2,3$) are F-saturated, yielding epimorphic-closedness in those cases. A key finding is that the smallest F-epimorphically closed pseudovariety containing the semigroup $Y$ is the full semigroup pseudovariety $\mathsf{S}$, and a complete classification is achieved: any F-saturated pseudovariety lies within one of the $\mathsf{V}_i$, and otherwise no proper F-epimorphically closed pseudovariety can contain it. The paper also provides constructive methods to reach completely 0-simple and then all finite semigroups from $Y$-based building blocks, clarifying the reach of epimorphic closure in this setting and highlighting decidability consequences tied to $Y$-membership.

Abstract

For each of the following conditions, we characterize the pseudovarieties of semigroups V that satisfy it: (i) every epimorphism to a member of V is onto; (ii) every epimorphism to a finite semigroup with domain a member of V is onto; (iii) for every epimorphism from S to T with S in V and T finite, T is also a member of V.

Figures (2)

• Figure 1: A sketch of the eggbox picture of the nonzero $\mathcal{D}$-class of the semigroup $T$ in the proof of Theorem \ref{['t:from-Y-to-CS0']}.
• Figure 2: The automaton $\tilde{\mathcal{A}}$, where $\mathcal{A}$ is drawn in thick lines and $T(\mathcal{A})=T_4$.

Theorems & Definitions (39)

• Theorem 1.1: Isbell's Zigzag Theorem
• Lemma 1.2
• proof
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Corollary 2.4