On the Monoid of Unital Endomorphisms of a Boolean Ring
Let <i>X</i> be a nonempty set and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)&l...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-11-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/10/4/305 |
| _version_ | 1797506573278380032 |
|---|---|
| author | Bana Al Subaiei Noômen Jarboui |
| author_facet | Bana Al Subaiei Noômen Jarboui |
| author_sort | Bana Al Subaiei |
| collection | DOAJ |
| description | Let <i>X</i> be a nonempty set and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> the power set of <i>X</i>. The aim of this paper is to provide an explicit description of the monoid <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>End</mi><msub><mn>1</mn><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msub></msub><mrow><mo>(</mo><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of unital ring endomorphisms of the Boolean ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the automorphism group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Aut</mi><mo>(</mo><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> when <i>X</i> is finite. Among other facts, it is shown that if <i>X</i> has cardinality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>End</mi><msub><mn>1</mn><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msub></msub><mrow><mo>(</mo><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≅</mo><msubsup><mi>T</mi><mi>n</mi><mrow><mi>o</mi><mi>p</mi></mrow></msubsup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>n</mi></msub></semantics></math></inline-formula> is the full transformation monoid on the set <i>X</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Aut</mi><mrow><mo>(</mo><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≅</mo><msub><mi>S</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T04:34:27Z |
| format | Article |
| id | doaj.art-29dc70f9c9a84926bb4f18b27eff98b0 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T04:34:27Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-29dc70f9c9a84926bb4f18b27eff98b02023-11-23T03:49:49ZengMDPI AGAxioms2075-16802021-11-0110430510.3390/axioms10040305On the Monoid of Unital Endomorphisms of a Boolean RingBana Al Subaiei0Noômen Jarboui1Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi ArabiaDépartement de Mathématiques, Faculté des Sciences de Sfax, Université de Sfax, P.O. Box 1171, Route de Soukra, Sfax 3038, TunisiaLet <i>X</i> be a nonempty set and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> the power set of <i>X</i>. The aim of this paper is to provide an explicit description of the monoid <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>End</mi><msub><mn>1</mn><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msub></msub><mrow><mo>(</mo><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of unital ring endomorphisms of the Boolean ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the automorphism group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Aut</mi><mo>(</mo><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> when <i>X</i> is finite. Among other facts, it is shown that if <i>X</i> has cardinality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>End</mi><msub><mn>1</mn><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msub></msub><mrow><mo>(</mo><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≅</mo><msubsup><mi>T</mi><mi>n</mi><mrow><mi>o</mi><mi>p</mi></mrow></msubsup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>n</mi></msub></semantics></math></inline-formula> is the full transformation monoid on the set <i>X</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Aut</mi><mrow><mo>(</mo><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≅</mo><msub><mi>S</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/10/4/305Boolean ringpower setprime idealmaximal idealring endomorphism |
| spellingShingle | Bana Al Subaiei Noômen Jarboui On the Monoid of Unital Endomorphisms of a Boolean Ring Axioms Boolean ring power set prime ideal maximal ideal ring endomorphism |
| title | On the Monoid of Unital Endomorphisms of a Boolean Ring |
| title_full | On the Monoid of Unital Endomorphisms of a Boolean Ring |
| title_fullStr | On the Monoid of Unital Endomorphisms of a Boolean Ring |
| title_full_unstemmed | On the Monoid of Unital Endomorphisms of a Boolean Ring |
| title_short | On the Monoid of Unital Endomorphisms of a Boolean Ring |
| title_sort | on the monoid of unital endomorphisms of a boolean ring |
| topic | Boolean ring power set prime ideal maximal ideal ring endomorphism |
| url | https://www.mdpi.com/2075-1680/10/4/305 |
| work_keys_str_mv | AT banaalsubaiei onthemonoidofunitalendomorphismsofabooleanring AT noomenjarboui onthemonoidofunitalendomorphismsofabooleanring |