On monoids of injective partial cofinite selfmaps

Abstract

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