On monoids of injective partial selfmaps almost everywhere the identity

Abstract

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

arXiv comment16 pages