On the semigroup of injective transformations with restricted range that equal gap and defect
Let \(X\) be an infinite set and \(I(X)\) the symmetric inverse semigroup on \(X\). Let \(A(X)=\{\alpha \in I(X):|X\setminus \mathrm{dom\;}\alpha|=|X\setminus X\alpha|\}\), it is known that \(A(X)\) is the largest factorizable subsemigroup of \(I(X)\). In this article, for any nonempty subset \(Y\)...
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| Date: | 2025 |
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Lugansk National Taras Shevchenko University
2025
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admjournalluguniveduua-article-23352025-04-13T15:32:01Z On the semigroup of injective transformations with restricted range that equal gap and defect Singha, Boorapa transformation semigroup, Green's relation, natural partial order 20M20 Let \(X\) be an infinite set and \(I(X)\) the symmetric inverse semigroup on \(X\). Let \(A(X)=\{\alpha \in I(X):|X\setminus \mathrm{dom\;}\alpha|=|X\setminus X\alpha|\}\), it is known that \(A(X)\) is the largest factorizable subsemigroup of \(I(X)\). In this article, for any nonempty subset \(Y\) of \(X\), we consider the subsemigroup \(A(X, Y)\) of \(A(X)\) of all transformations with range contained in \(Y\). We give a complete description of Green's relations on \(A(X,Y)\). With respect to the natural partial order on a semigroup, we determine when two elements in \(A(X,Y)\) are related and find all the maximum, minimum, maximal, minimal, lower cover and upper cover elements. We also describe elements which are compatible and we investigate the greatest lower bound and the least upper bound of two elements in \(A(X,Y)\). Lugansk National Taras Shevchenko University Thailand Science Research and Innovation (TSRI) Chiang Mai Rajabhat University 2025-04-13 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2335 10.12958/adm2335 Algebra and Discrete Mathematics; Vol 39, No 1 (2025) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2335/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2335/1255 Copyright (c) 2025 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2025-04-13T15:32:01Z |
| collection |
OJS |
| language |
English |
| topic |
transformation semigroup Green's relation natural partial order 20M20 |
| spellingShingle |
transformation semigroup Green's relation natural partial order 20M20 Singha, Boorapa On the semigroup of injective transformations with restricted range that equal gap and defect |
| topic_facet |
transformation semigroup Green's relation natural partial order 20M20 |
| format |
Article |
| author |
Singha, Boorapa |
| author_facet |
Singha, Boorapa |
| author_sort |
Singha, Boorapa |
| title |
On the semigroup of injective transformations with restricted range that equal gap and defect |
| title_short |
On the semigroup of injective transformations with restricted range that equal gap and defect |
| title_full |
On the semigroup of injective transformations with restricted range that equal gap and defect |
| title_fullStr |
On the semigroup of injective transformations with restricted range that equal gap and defect |
| title_full_unstemmed |
On the semigroup of injective transformations with restricted range that equal gap and defect |
| title_sort |
on the semigroup of injective transformations with restricted range that equal gap and defect |
| description |
Let \(X\) be an infinite set and \(I(X)\) the symmetric inverse semigroup on \(X\). Let \(A(X)=\{\alpha \in I(X):|X\setminus \mathrm{dom\;}\alpha|=|X\setminus X\alpha|\}\), it is known that \(A(X)\) is the largest factorizable subsemigroup of \(I(X)\). In this article, for any nonempty subset \(Y\) of \(X\), we consider the subsemigroup \(A(X, Y)\) of \(A(X)\) of all transformations with range contained in \(Y\). We give a complete description of Green's relations on \(A(X,Y)\). With respect to the natural partial order on a semigroup, we determine when two elements in \(A(X,Y)\) are related and find all the maximum, minimum, maximal, minimal, lower cover and upper cover elements. We also describe elements which are compatible and we investigate the greatest lower bound and the least upper bound of two elements in \(A(X,Y)\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2025 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2335 |
| work_keys_str_mv |
AT singhaboorapa onthesemigroupofinjectivetransformationswithrestrictedrangethatequalgapanddefect |
| first_indexed |
2025-12-02T15:39:46Z |
| last_indexed |
2025-12-02T15:39:46Z |
| _version_ |
1850412151242489856 |