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.
Categories: math.GR