Consider the following generalization of the bicyclic monoid. Let $\kappa$ be any infinite cardinal and let $\mathcal{ IP\!F}\left(\sigma{ \mathbb{ N}^\kappa}\right)$ 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 $\mathcal{ IP\!F}\left(\sigma{ \mathbb{ N}^\kappa}\right)$, show that it is bisimple, $E$-unitary, $F$-inverse semigroup, describe Green's relations on $\mathcal{ IP\!F}\left(\sigma{ \mathbb{ N}^\kappa}\right)$, describe the group of units $H\left(\mathbb{ I}\right)$ of the semigroup $\mathcal{ IP\!F}\left(\sigma{ \mathbb{ N}^\kappa}\right)$ and describe its maximal subgroups. We prove that the semigroup $\mathcal{ IP\!F}\left(\sigma{ \mathbb{ N}^\kappa}\right)$ is isomorphic to the semidirect product $\mathcal{ S}_\kappa\ltimes\sigma{ \mathbb{ B}^\kappa}$ of the semigroup $\sigma{ \mathbb{ B}^\kappa}$ by the group $\mathcal{ S}_\kappa$, show that every non-identity congruence $\mathfrak{ C}$ on the semigroup $\mathcal{ IP\!F}\left(\sigma{ \mathbb{ N}^\kappa}\right)$ is a group congruence and describe the least group congruence on $\mathcal{ IP\!F}\left(\sigma{ \mathbb{ N}^\kappa}\right)$.
Categories: math.GR