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

Research paper by **Oleg Gutik, Inna Pozdnyakova**

Indexed on: **29 Jun '14**Published on: **29 Jun '14**Published in: **Mathematics - Group Theory**

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#### 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 $S$.