On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images

Abstract

We study the semigroup \(\mathscr{I\!O}\!_{\infty} (\mathbb{Z}^n_{\operatorname{lex}})\) of monotone injective partial selfmaps of the set of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) having co-finite domain and image, where \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) is the lexicographic product of \(n\)-elements chain and the set of integers with the usual order.  We show that \(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})\) is bisimple and establish its projective congruences. We prove that \(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})\) is finitely generated, and for \(n=1\) every automorphism of \(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})\) is inner and show that  in the case \(n \geqslant 2\) the semigroup \(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})\) has non-inner automorphisms. Also we show that every Baire topology \(\tau\) on \(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})\) such that \((\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}}),\tau)\) is a Hausdorff semitopological semigroup is discrete, construct a non-discrete Hausdorff semigroup inverse topology on \(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})\), and prove that the discrete semigroup \(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})\) 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 \(S\).