Monoids of injective maps closed under conjugation by permutations

TL;DR

This paper classifies all submonoids of the monoid of injective maps on a countably infinite set that remain invariant under conjugation by permutations, providing a comprehensive structural understanding.

Contribution

It provides a complete classification of submonoids of Inj(X) closed under conjugation by Sym(X), a novel structural insight in the theory of injective maps.

Findings

Complete classification of conjugation-closed submonoids of Inj(X)

Structural characterization of these submonoids

Extension of symmetry concepts in monoid theory

Abstract

Let X be a countably infinite set, Inj(X) the monoid of all injective endomaps of X, and Sym(X) the group of all permutations of X. We classify all submonoids of Inj(X) that are closed under conjugation by elements of Sym(X).