@article{Gutik_Savchuk_2019, title={On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers: Array}, volume={11}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/2109}, DOI={10.15330/cmp.11.2.296-310}, abstractNote={<p>In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\! earrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{ earrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\! earrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N }$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\! earrow}(\mathbb{N})$ 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 $\mathscr{I}_{\infty}^{\! earrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N }$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\! earrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\! earrow}(\mathbb{N})$ 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 $\mathscr{I}_{\infty}^{\,\Rsh\!\! earrow}(\mathbb{N})$ 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 the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\! earrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N }$ and embeddings of such semigroups into compact-like topological semigroups.</p>}, number={2}, journal={Carpathian Mathematical Publications}, author={Gutik, O.V. and Savchuk, A.S.}, year={2019}, month={Dec.}, pages={296–310} }