On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images
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 produc...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-10352018-04-26T02:11:00Z On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images Gutik, Oleg Pozdnyakova, Inna Topological semigroup, semitopological semigroup, semigroup of bijective partial transformations, symmetric inverse semigroup, congruence, ideal, automorphism, homomorphism, Baire space, semigroup topologization, embedding 20M18, 20M20; 20M05, 20M15, 22A15, 54C25, 54D40, 54E52, 54H10 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 \(S\). Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1035 Algebra and Discrete Mathematics; Vol 17, No 2 (2014) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1035/558 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Topological semigroup semitopological semigroup semigroup of bijective partial transformations symmetric inverse semigroup congruence ideal automorphism homomorphism Baire space semigroup topologization embedding 20M18 20M20; 20M05 20M15 22A15 54C25 54D40 54E52 54H10 |
| spellingShingle |
Topological semigroup semitopological semigroup semigroup of bijective partial transformations symmetric inverse semigroup congruence ideal automorphism homomorphism Baire space semigroup topologization embedding 20M18 20M20; 20M05 20M15 22A15 54C25 54D40 54E52 54H10 Gutik, Oleg Pozdnyakova, Inna On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images |
| topic_facet |
Topological semigroup semitopological semigroup semigroup of bijective partial transformations symmetric inverse semigroup congruence ideal automorphism homomorphism Baire space semigroup topologization embedding 20M18 20M20; 20M05 20M15 22A15 54C25 54D40 54E52 54H10 |
| format |
Article |
| author |
Gutik, Oleg Pozdnyakova, Inna |
| author_facet |
Gutik, Oleg Pozdnyakova, Inna |
| author_sort |
Gutik, Oleg |
| title |
On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images |
| title_short |
On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images |
| title_full |
On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images |
| title_fullStr |
On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images |
| title_full_unstemmed |
On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images |
| title_sort |
on monoids of monotone injective partial selfmaps of \(l_n\times_{\operatorname{lex}}\mathbb{z}\) with co-finite domains and images |
| description |
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 \(S\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1035 |
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2024-04-12T06:25:58Z |
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2024-04-12T06:25:58Z |
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