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

Abstract

In this paper we study submonoids of the monoid ${\mathcal{I}}_{\infty }^{\Rsh \nearrow }(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let ${\mathcal{I}}_{\infty }^{\nearrow }(\mathbb{N})$ be a submonoid of ${\mathcal{I}}_{\infty }^{\Rsh \nearrow }(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and ${\mathcal{C}}_{\mathbb{N}}$ be a subsemigroup ${\mathcal{I}}_{\infty }^{\Rsh \nearrow }(\mathbb{N})$ 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 ${\mathcal{I}}_{\infty }^{\nearrow }(\mathbb{N})$ which contains the semigroup ${\mathcal{C}}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}{\mathbb{N}}_{\infty }^{[\underline{1}]}$ of ${\mathcal{I}}_{\infty }^{\Rsh \nearrow }(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of ${\mathcal{I}}_{\infty }^{\Rsh \nearrow }(\mathbb{N})$ 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 ${\mathcal{I}}_{\infty }^{\Rsh \nearrow }(\mathbb{N})$ such that $S$ contains ${\mathcal{C}}_{\mathbb{N}}$ 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 ${\mathcal{I}}_{\infty }^{\Rsh \nearrow }(\mathbb{N})$ which contain ${\mathcal{C}}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

12 pages. arXiv admin note: text overlap with arXiv:1802.03598; and text overlap with arXiv:1908.08521 by other authors