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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2014, Volume 17, Issue 2, Pages 256–279 (Mi adm470)

RESEARCH ARTICLE

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

Oleg Gutik, Inna Pozdnyakova

Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv, Universytetska 1, Lviv, 79000, Ukraine

Abstract: We study the semigroup $\mathscr{IO}_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ 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 $\mathscr{IO}_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is bisimple and establish its projective congruences. We prove that $\mathscr{IO}_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is finitely generated, and for $n=1$ every automorphism of $\mathscr{IO}_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is inner and show that in the case $n\geqslant 2$ the semigroup $\mathscr{IO}_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ has non-inner automorphisms. Also we show that every Baire topology $\tau$ on $\mathscr{IO}_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ such that $(\mathscr{IO}_{\infty}(\mathbb{Z}^n_{\operatorname{lex}}),\tau)$ is a Hausdorff semitopological semigroup is discrete, construct a non-discrete Hausdorff semigroup inverse topology on $\mathscr{IO}_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$, and prove that the discrete semigroup $\mathscr{IO}_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ 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 $S$.

Keywords: topological semigroup, semitopological semigroup, semigroup of bijective partial transformations, symmetric inverse semigroup, congruence, ideal, automorphism, homomorphism, Baire space, semigroup topologization, embedding.

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Bibliographic databases:
MSC: Primary 20M18, 20M20; Secondary 20M05, 20M15, 22A15, 54C25, 54D40, 54E52, 54H10
Revised: 27.01.2014
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Citation: Oleg Gutik, Inna Pozdnyakova, “On monoids of monotone injective partial selfmaps of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ with co-finite domains and images”, Algebra Discrete Math., 17:2 (2014), 256–279

Citation in format AMSBIB
\Bibitem{GutPoz14} \by Oleg~Gutik, Inna~Pozdnyakova \paper On monoids of monotone injective partial selfmaps of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ with co-finite domains and images \jour Algebra Discrete Math. \yr 2014 \vol 17 \issue 2 \pages 256--279 \mathnet{http://mi.mathnet.ru/adm470} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3287933}