On monoids of injective partial cofinite selfmaps

Abstract

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