Saturated (<i>n</i>, <i>m</i>)-Regular Semigroups
The aim of this paper is to determine several saturated classes of structurally regular semigroups. First, we show that structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo>&l...
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MDPI AG
2023-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/9/2203 |
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| author | Amal S. Alali Sakeena Bano Muneer Nabi |
| author_facet | Amal S. Alali Sakeena Bano Muneer Nabi |
| author_sort | Amal S. Alali |
| collection | DOAJ |
| description | The aim of this paper is to determine several saturated classes of structurally regular semigroups. First, we show that structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula>-regular semigroups are saturated in a subclass of semigroups for any pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula> of positive integers. We also demonstrate that, for all positive integers <i>n</i> and <i>k</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>, the variety of structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-left seminormal bands is saturated in the variety of structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-bands. As a result, in the category of structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-bands, epis from structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-left seminormal bands is onto. |
| first_indexed | 2024-03-11T04:12:39Z |
| format | Article |
| id | doaj.art-f42251e9fa754f3bac392e5b3695a3d5 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:12:39Z |
| publishDate | 2023-05-01 |
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| spelling | doaj.art-f42251e9fa754f3bac392e5b3695a3d52023-11-17T23:21:21ZengMDPI AGMathematics2227-73902023-05-01119220310.3390/math11092203Saturated (<i>n</i>, <i>m</i>)-Regular SemigroupsAmal S. Alali0Sakeena Bano1Muneer Nabi2Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaDepartment of Mathematics, Central University of Kashmir, Ganderbal 191201, IndiaDepartment of Mathematics, Central University of Kashmir, Ganderbal 191201, IndiaThe aim of this paper is to determine several saturated classes of structurally regular semigroups. First, we show that structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula>-regular semigroups are saturated in a subclass of semigroups for any pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula> of positive integers. We also demonstrate that, for all positive integers <i>n</i> and <i>k</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>, the variety of structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-left seminormal bands is saturated in the variety of structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-bands. As a result, in the category of structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-bands, epis from structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-left seminormal bands is onto.https://www.mdpi.com/2227-7390/11/9/2203dominionsepimorphismszigzagsaturatedstructurally regular |
| spellingShingle | Amal S. Alali Sakeena Bano Muneer Nabi Saturated (<i>n</i>, <i>m</i>)-Regular Semigroups Mathematics dominions epimorphisms zigzag saturated structurally regular |
| title | Saturated (<i>n</i>, <i>m</i>)-Regular Semigroups |
| title_full | Saturated (<i>n</i>, <i>m</i>)-Regular Semigroups |
| title_fullStr | Saturated (<i>n</i>, <i>m</i>)-Regular Semigroups |
| title_full_unstemmed | Saturated (<i>n</i>, <i>m</i>)-Regular Semigroups |
| title_short | Saturated (<i>n</i>, <i>m</i>)-Regular Semigroups |
| title_sort | saturated i n i i m i regular semigroups |
| topic | dominions epimorphisms zigzag saturated structurally regular |
| url | https://www.mdpi.com/2227-7390/11/9/2203 |
| work_keys_str_mv | AT amalsalali saturatediniimiregularsemigroups AT sakeenabano saturatediniimiregularsemigroups AT muneernabi saturatediniimiregularsemigroups |