The monoid of order isomorphisms of principal filters of a power of the positive integers

Abstract

Let $n$ be any positive integer and $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We study algebraic properties of the semigroup $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$. In particular, we show that $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$ is a bisimple, $E$-unitary, $F$-inverse semigroup, describe Green's relations on $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$ and its maximal subgroups. We show that the semigroup $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$ is isomorphic to the semidirect product of the direct $n$-th power of the bicyclic monoid ${\mathcal{C}}^n(p,q)$ by the group of permutation ${\mathcal{S}}_n$. Also we prove that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$ is group and describe the least group congruence on $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$. We show that every Hausdorff shift-continuous topology on $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$ is discrete and discuss embedding of the semigroup $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$ into compact-like topological semigroups.

17 pages