# On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers

## TL;DR

This paper investigates the algebraic structure and automorphisms of certain inverse submonoids of almost monotone injective co-finite partial selfmaps of positive integers, revealing their simplicity and connection to the additive group of integers.

## Contribution

It characterizes automorphisms, congruences, and topological embeddings of specific inverse submonoids within the monoid of almost monotone partial selfmaps of positive integers.

## Key findings

- Automorphisms of certain inverse subsemigroups are trivial (identity).
- Inverse submonoids containing specific subsemigroups are simple and have quotients isomorphic to integers.
- Studied topologizations and embeddings into compact-like topological semigroups.

## Abstract

In this paper we study submonoids of the monoid $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_\infty^{\!\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_\infty^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_\infty^{[\underline{1}]}$ of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_\infty^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.