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Algebra and Discrete Mathematics, 2014, Volume 17, Issue 2, Pages 256–279
(Mi adm470)
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This article is cited in 3 scientific papers (total in 3 papers)
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 $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ 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 $S$.
Keywords:
topological semigroup, semitopological semigroup, semigroup of bijective partial transformations, symmetric inverse semigroup, congruence, ideal, automorphism, homomorphism, Baire space, semigroup topologization, embedding.
Received: 07.12.2013 Revised: 27.01.2014
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
Linking options:
https://www.mathnet.ru/eng/adm470 https://www.mathnet.ru/eng/adm/v17/i2/p256
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Abstract page: | 246 | Full-text PDF : | 87 | References: | 45 |
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