The monoid of order isomorphisms between principal filters of $σ{\mathbb{N}^κ}$

Abstract

Consider the following generalization of the bicyclic monoid. Let $κ$ be any infinite cardinal and let $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$ be the semigroup of all order isomorphisms between principal filters of the set $σ{\mathbb{N}^κ}$ with the product order. We shall study algebraic properties of the semigroup $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$, show that it is bisimple, $E$-unitary, $F$-inverse semigroup, describe Green's relations on $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$, describe the group of units $H\left(\mathbb{I}\right)$ of the semigroup $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$ and describe its maximal subgroups. We prove that the semigroup $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$ is isomorphic to the semidirect product $\mathcal{S}_κ\ltimesσ{\mathbb{B}^κ}$ of the semigroup $σ{\mathbb{B}^κ}$ by the group $\mathcal{S}_κ$, show that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$ is a group congruence and describe the least group congruence on $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$.