On monoids of monotone injective partial selfmaps of $L_n{\times }_{\mathrm{lex}}\mathbb{Z}$ with co-finite domains and images

We study the semigroup $\mathcal{I}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{Z}}_{\mathrm{lex}}^{})$ of monotone injective partial selfmaps of the set of $L_n{\times }_{\mathrm{lex}}\mathbb{Z}$ having co-finite domain and image, where $L_n{\times }_{\mathrm{lex}}\mathbb{Z}$ is the lexicographic product of $n$-elements chain and the set of integers with the usual order. We show that $\mathcal{I}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{Z}}_{\mathrm{lex}}^{})$ is bisimple and establish its projective congruences. We prove that $\mathcal{I}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{Z}}_{\mathrm{lex}}^{})$ is finitely generated, and for $n=1$ every automorphism of $\mathcal{I}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{Z}}_{\mathrm{lex}}^{})$ is inner and show that in the case $n⩾2$ the semigroup $\mathcal{I}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{Z}}_{\mathrm{lex}}^{})$ has non-inner automorphisms. Also we show that every Baire topology $\tau$ on $\mathcal{I}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{Z}}_{\mathrm{lex}}^{})$ such that $(\mathcal{I}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{Z}}_{\mathrm{lex}}^{}),\tau )$ is a Hausdorff semitopological semigroup is discrete, construct a non-discrete Hausdorff semigroup inverse topology on $\mathcal{I}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{Z}}_{\mathrm{lex}}^{})$, and prove that the discrete semigroup $\mathcal{I}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{Z}}_{\mathrm{lex}}^{})$ cannot be embedded into some classes of compact-like topological semigroups and that its remainder under the closure in a topological semigroup $S$ is an ideal in $S$.