Presentations for monoids of endomorphisms of a star graph

TL;DR

The paper determines explicit presentations for the monoids of endomorphisms, strong/weak endomorphisms, and strong weak endomorphisms of the star graph $S_n$, building on standard monoid-presentation techniques. It leverages a canonical-form/Guess-and-Prove approach to extend a known presentation of $\mathscr{T}^0_{n-1}$ with additional generators $z$, $z_0$, and $c_0$ to obtain presentations for End $S_n$, swEnd $S_n$, and wEnd $S_n$, respectively, and proves the correctness via canonical forms. It also provides cardinalities, regularity, ranks of generating sets, and explicit small-$n$ presentations (with GAP verification), contributing concrete algebraic descriptions of these graph-endomorphism monoids. These results facilitate computational and theoretical analysis of endomorphism monoids of star graphs and have potential applications in automata theory and related algebraic graph theory contexts.

Abstract

In this paper, we consider the monoids of all endomorphisms, of all weak endomorphisms, of all strong endomorphisms and of all 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.

Theorems & Definitions (12)

• Proposition 1.1
• Theorem 1.2: Guess and Prove method
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof