The monoid of order isomorphisms between principal filters of $\sigma {\mathbb{N}}^{\kappa }$

Consider the following generalization of the bicyclic monoid. Let $\kappa$ be any infinite cardinal and let $\mathcal{I}\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{F}\left(\sigma {\mathbb{N}}^{\kappa }\right)$ be the semigroup of all order isomorphisms between principal filters of the set $\sigma {\mathbb{N}}^{\kappa }$ with the product order. We shall study algebraic properties of the semigroup $\mathcal{I}\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{F}\left(\sigma {\mathbb{N}}^{\kappa }\right)$, show that it is bisimple, $E$-unitary, $F$-inverse semigroup, describe Green’s relations on $\mathcal{I}\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{F}\left(\sigma {\mathbb{N}}^{\kappa }\right)$, describe the group of units $H\left(\mathbb{I}\right)$ of the semigroup $\mathcal{I}\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{F}\left(\sigma {\mathbb{N}}^{\kappa }\right)$ and describe its maximal subgroups. We prove that the semigroup $\mathcal{I}\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{F}\left(\sigma {\mathbb{N}}^{\kappa }\right)$ is isomorphic to the semidirect product ${\mathcal{S}}_{\kappa }⋉\sigma {\mathbb{B}}^{\kappa }$ of the semigroup $\sigma {\mathbb{B}}^{\kappa }$ by the group ${\mathcal{S}}_{\kappa }$, show that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathcal{I}\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{F}\left(\sigma {\mathbb{N}}^{\kappa }\right)$ is a group congruence and describe the least group congruence on $\mathcal{I}\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{F}\left(\sigma {\mathbb{N}}^{\kappa }\right)$.