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

Research paper by **Oleg Gutik, Anatolii Savchuk**

Indexed on: **29 Apr '19**Published on: **26 Apr '19**Published in: **arXiv - Mathematics - Group Theory**

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#### 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.