On monoids of injective partial cofinite selfmaps

TL;DR

This paper investigates the algebraic structure of the semigroup of injective partial cofinite selfmaps on an infinite set, revealing its bisimplicity, Green relations, and the nature of its congruences.

Contribution

It provides a detailed structural analysis of the semigroup, including its bisimplicity, idempotent chains, Green relations, and the characterization of its group congruences.

Findings

The semigroup is bisimple and inverse.

Every non-trivial congruence is a group congruence.

The structure of the quotient semigroup is characterized.

Abstract

We study the semigroup $\mathscr{I}^{\mathrm{cf}}_\lambda$ of injective partial cofinite selfmaps of an infinite cardinal $\lambda$. We show that $\mathscr{I}^{\mathrm{cf}}_\lambda$ is a bisimple inverse semigroup and each chain of idempotents in $\mathscr{I}^{\mathrm{cf}}_\lambda$ is contained in a bicyclic subsemigroup of $\mathscr{I}^{\mathrm{cf}}_\lambda$, we describe the Green relations on $\mathscr{I}^{\mathrm{cf}}_\lambda$ and we prove that every non-trivial congruence on $\mathscr{I}^{\mathrm{cf}}_\lambda$ is a group congruence. Also, we describe the structure of the quotient semigroup $\mathscr{I}^{\mathrm{cf}}_\lambda/\sigma$, where $\sigma$ is the least group congruence on $\mathscr{I}^{\mathrm{cf}}_\lambda$.