On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers
In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nea...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2019-12-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/2109 |
| _version_ | 1818310665959374848 |
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| author | O.V. Gutik A.S. Savchuk |
| author_facet | O.V. Gutik A.S. Savchuk |
| author_sort | O.V. Gutik |
| collection | DOAJ |
| description | In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups. |
| first_indexed | 2024-12-13T07:49:41Z |
| format | Article |
| id | doaj.art-42cbb690d4e7416394fc7ba2938596f4 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-12-13T07:49:41Z |
| publishDate | 2019-12-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-42cbb690d4e7416394fc7ba2938596f42022-12-21T23:54:44ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102019-12-0111229631010.15330/cmp.11.2.296-3102109On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integersO.V. Gutik0A.S. Savchuk1Ivan Franko National University, 1 Universytetska str., 79000, Lviv, UkraineIvan Franko National University, 1 Universytetska str., 79000, Lviv, UkraineIn this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.https://journals.pnu.edu.ua/index.php/cmp/article/view/2109inverse semigroupisometrypartial bijectioncongruencebicyclic semigroupsemitopological semigrouptopological semigroupdiscrete topologyembeddingbohr compactification |
| spellingShingle | O.V. Gutik A.S. Savchuk On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers Karpatsʹkì Matematičnì Publìkacìï inverse semigroup isometry partial bijection congruence bicyclic semigroup semitopological semigroup topological semigroup discrete topology embedding bohr compactification |
| title | On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers |
| title_full | On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers |
| title_fullStr | On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers |
| title_full_unstemmed | On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers |
| title_short | On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers |
| title_sort | on inverse submonoids of the monoid of almost monotone injective co finite partial selfmaps of positive integers |
| topic | inverse semigroup isometry partial bijection congruence bicyclic semigroup semitopological semigroup topological semigroup discrete topology embedding bohr compactification |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/2109 |
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