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

TL;DR

This paper investigates the algebraic structure of the semigroup of order isomorphisms between principal filters of the n-th power of positive integers, revealing its properties, isomorphisms, and topological characteristics.

Contribution

It characterizes the algebraic and topological properties of the semigroup of order isomorphisms of principal filters in the n-dimensional positive integer lattice.

Findings

The semigroup is bisimple, E-unitary, F-inverse.

It is isomorphic to a semidirect product involving the bicyclic monoid and permutation group.

Every non-identity congruence is a group congruence.

Abstract

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