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Irreducibility of Semigroup Morphisms

Paul C. Bell, Eva Foster, Daniel Reidenbach

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.

Irreducibility of Semigroup Morphisms

Abstract

We study the notion of irreducibility of semigroup morphisms. Given an alphabet , a morphism is irreducible if any factorisation can only be satisfied if or is a trivial morphism. Otherwise, is reducible. We introduce the notion of irreducibility, characterise this property and study a number of fundamental questions on the concepts under consideration.
Paper Structure (9 sections, 26 theorems, 1 figure, 2 tables)

This paper contains 9 sections, 26 theorems, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Observations and Fundamental Characterisations
  4. Uniqueness of Factorisations
  5. Incidence Matrices
  8. Appendix: Exhaustive Search
  9. Appendix: Derivation graph

Key Result

proposition 1

Let $\varphi:\Sigma^+\rightarrow\Sigma^+$ be a morphism for $\Sigma=\{a_1, a_2, \dots, a_n\}$ and define $W = \{\varphi(a_i) \mid 1 \leq i \leq n\}$. If $|\mathop{\mathrm{symb}}\nolimits(W)| < n$, then $\varphi$ is reducible.

Figures (1)

  • Figure 1: The derivation graph of $\varphi$, as given in Example \ref{['ex:factorisation tree']}

Theorems & Definitions (54)

  • definition 1
  • proposition 1
  • proof
  • proposition 2
  • proof
  • definition 2
  • theorem 1
  • proof
  • definition 3
  • corollary 1
  • ...and 44 more