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On 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 determine their ranks. We also describe their Green's relations, calculate their cardinalities and study their regularity.

On 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 determine their ranks. We also describe their Green's relations, calculate their cardinalities and study their regularity.
Paper Structure (3 sections, 31 theorems, 21 equations)

This paper contains 3 sections, 31 theorems, 21 equations.

Table of Contents

  1. Cardinalities
  2. Regularity and Green's relations
  3. Generators and ranks

Key Result

Proposition 1.1

For an integer $n\geqslant1$, let $\alpha \in \mathcal{PT}(\Omega_{n-1}^0)$. Then, $\alpha\in\mathrm{PwEnd}(S_n)$ if and only if one of the following (mutually disjoint) conditions holds:

Theorems & Definitions (48)

  • Proposition 1.1
  • Corollary 1.2
  • proof
  • Corollary 1.3
  • Proposition 1.4
  • Corollary 1.5
  • Proposition 1.6
  • Corollary 1.7
  • Proposition 1.8
  • Corollary 1.9
  • ...and 38 more