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Presentations for monoids of partial endomorphisms of a star graph

Ilinka Dimitrova, Vítor H. Fernandes, Jörg Koppitz

Abstract

In this paper, we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.

Presentations for monoids of partial endomorphisms of a star graph

Abstract

In this paper, we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.

Paper Structure

This paper contains 7 sections, 25 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. On partial endomorphisms of a star graph
  3. A presentation for $\mathrm{PsEnd}(S_n)$
  4. A presentation for $\mathrm{PswEnd}(S_n)$
  5. A presentation for $\mathrm{PEnd}(S_n)$
  6. A presentation for $\mathrm{PwEnd}(S_n)$
  7. A presentation for $\mathrm{IEnd}(S_n)$

Key Result

Proposition 1.1

Let $M$ be a monoid generated by a set $X$. Then, $\langle X\mid R\rangle$ is a presentation for $M$ if and only if the following two conditions are satisfied:

Theorems & Definitions (38)

  • Proposition 1.1
  • Theorem 1.2: Guess and Prove method
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Lemma 4.1
  • ...and 28 more