Irreducibility of Semigroup Morphisms

2026-03-16

AI Summary

This study addresses the irreducibility of semigroup homomorphisms—specifically, whether a given homomorphism cannot be decomposed into a composition of two nontrivial homomorphisms. The paper presents the first formalization of this notion, establishing rigorous criteria and a theoretical framework for characterizing irreducible homomorphisms. By integrating techniques from algebraic structure analysis, formal language theory, and homomorphism decomposition, the work provides a systematic characterization of such irreducible mappings. This contribution not only fills a notable gap in the theory of semigroups concerning homomorphism decomposition but also offers novel perspectives and tools for automata theory and related algebraic problems.

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• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Natural Language Processing: Lexical Semantics and Morphology
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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Abstract

We study the notion of irreducibility of semigroup morphisms. Given an alphabet $Σ$, a morphism $\varphi:Σ^+\rightarrowΣ^+$ is irreducible if any factorisation $\varphi=ψ_2\circψ_1$ can only be satisfied if $ψ_1$ or $ψ_2$ is a trivial morphism. Otherwise, $\varphi$ is reducible. We introduce the notion of irreducibility, characterise this property and study a number of fundamental questions on the concepts under consideration.

Problem

• irreducibility
• semigroup morphisms
• factorisation
• trivial morphism
• alphabet

Innovation

• irreducibility
• semigroup morphisms
• factorisation
• formal languages
• algebraic structures