Congruences on the monoid of monotone injective partial selfmaps of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ with co-finite domains and images

TL;DR

This paper investigates the structure of congruences in a semigroup of monotone injective partial selfmaps with co-finite domains and images on a lexicographically ordered set, focusing on those containing the least group congruence.

Contribution

It characterizes the sublattice of congruences containing the least group congruence in a specific semigroup of partial selfmaps.

Findings

Describes the sublattice of congruences containing the least group congruence.

Provides structural insights into the semigroup of monotone injective partial selfmaps.

Analyzes congruences on the semigroup of co-finite domain and image maps.

Abstract

We study congruences of the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ of monotone injective partial selfmaps of the set of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ having co-finite domain and image, where $L_n\times_{\operatorname{lex}}\mathbb{Z}$ is the lexicographic product of $n$-elements chain and the set of integers with the usual linear order. The structure of the sublattice of congruences on $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ which contain in the least group congruence is described.