On monoids of injective partial selfmaps almost everywhere the identity

TL;DR

This paper investigates the algebraic and topological properties of the semigroup of injective partial selfmaps that are identity almost everywhere on infinite sets, including its Green relations, ideals, congruences, and possible topologies.

Contribution

It characterizes the algebraic structure of the semigroup and explores the topological constraints, including the non-existence of certain embeddings and the construction of non-discrete topologies.

Findings

Green relations and ideals are fully described.

Every Hausdorff hereditary Baire topology on the semigroup is discrete.

The semigroup cannot embed into a compact topological semigroup.

Abstract

In this paper we study the semigroup $\mathscr{I}^{\infty}_\lambda$ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality $\lambda$. We describe the Green relations on $\mathscr{I}^{\infty}_\lambda$, all (two-sided) ideals and all congruences of the semigroup $\mathscr{I}^{\infty}_\lambda$. We prove that every Hausdorff hereditary Baire topology $\tau$ on $\mathscr{I}^{\infty}_\omega$ such that $(\mathscr{I}^{\infty}_\omega,\tau)$ is a semitopological semigroup is discrete and describe the closure of the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ in a topological semigroup. Also we show that for an infinite cardinal $\lambda$ the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning $\mathscr{I}^{\infty}_\lambda$ into a topological inverse semigroup.