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

TL;DR

This paper studies the algebraic structure of a semigroup of order isomorphisms between principal filters of a generalized set, revealing its properties, structure, and congruences, extending the bicyclic monoid concept to infinite cardinals.

Contribution

It introduces and analyzes the semigroup of order isomorphisms between principal filters of $\sigma{bN}^bk$, establishing its algebraic properties, structure, and congruence relations, generalizing known monoid concepts.

Findings

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

It is isomorphic to a semidirect product of a symmetric group and a semigroup of order-preserving transformations.

Every non-identity congruence is a group congruence.

Abstract

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 $\mathcal{IP\!F}\left(\sigma{\mathbb{N}^\kappa}\right)$.