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

• Дата: 2018
• Автори: Gutik, Oleg, Pozdnyakova, Inna
• Формат: Стаття
• Мова: English
• Опубліковано: Lugansk National Taras Shevchenko University 2018
• Теми: 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
• Онлайн доступ: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1035
• Назва журналу: Algebra and Discrete Mathematics

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 \(S\).