Algebra in superextensions of groups, I: zeros and commutativity
Given a group X we study the algebraic structure of its superextension λ(X). This is a right-topological semigroup consisting of all maximal linked systems on X
 
 endowed with the operation 
 
 
 A∘B={C⊂X:{x∈X:x−1C∈B}∈A}
 
 
 that...
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| Опубліковано в: : | Algebra and Discrete Mathematics |
|---|---|
| Дата: | 2008 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2008
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/153373 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Algebra in superextensions of groups, I: zeros and commutativity / T.T. Banakh, V. Gavrylkiv, O. Nykyforchyn // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 3. — С. 1–29. — Бібліогр.: 13 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862725040851648512 |
|---|---|
| author | Banakh, T.T. Gavrylkiv, V. Nykyforchyn, O. |
| author_facet | Banakh, T.T. Gavrylkiv, V. Nykyforchyn, O. |
| citation_txt | Algebra in superextensions of groups, I: zeros and commutativity / T.T. Banakh, V. Gavrylkiv, O. Nykyforchyn // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 3. — С. 1–29. — Бібліогр.: 13 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | Given a group X we study the algebraic structure of its superextension λ(X). This is a right-topological semigroup consisting of all maximal linked systems on X

endowed with the operation 


A∘B={C⊂X:{x∈X:x−1C∈B}∈A}


that extends the group operation of X. We characterize right zeros of λ(X) as invariant maximal linked systems on X and prove that λ(X) has a right zero if and only if each element of X has odd order. On the other hand, the semigroup λ(X) contains a left zero if and only if it contains a zero if and only if X has odd order |X|≤5. The semigroup λ(X) is commutative if and only if |X|≤4. We finish the paper with a complete description of the algebraic structure of the semigroups λ(X) for all groups X of cardinality |X|≤5.
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| first_indexed | 2025-12-07T18:50:59Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-153373 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T18:50:59Z |
| publishDate | 2008 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Banakh, T.T. Gavrylkiv, V. Nykyforchyn, O. 2019-06-14T03:39:46Z 2019-06-14T03:39:46Z 2008 Algebra in superextensions of groups, I: zeros and commutativity / T.T. Banakh, V. Gavrylkiv, O. Nykyforchyn // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 3. — С. 1–29. — Бібліогр.: 13 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20M99, 54B20. https://nasplib.isofts.kiev.ua/handle/123456789/153373 Given a group X we study the algebraic structure of its superextension λ(X). This is a right-topological semigroup consisting of all maximal linked systems on X
 
 endowed with the operation 
 
 
 A∘B={C⊂X:{x∈X:x−1C∈B}∈A}
 
 
 that extends the group operation of X. We characterize right zeros of λ(X) as invariant maximal linked systems on X and prove that λ(X) has a right zero if and only if each element of X has odd order. On the other hand, the semigroup λ(X) contains a left zero if and only if it contains a zero if and only if X has odd order |X|≤5. The semigroup λ(X) is commutative if and only if |X|≤4. We finish the paper with a complete description of the algebraic structure of the semigroups λ(X) for all groups X of cardinality |X|≤5. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Algebra in superextensions of groups, I: zeros and commutativity Article published earlier |
| spellingShingle | Algebra in superextensions of groups, I: zeros and commutativity Banakh, T.T. Gavrylkiv, V. Nykyforchyn, O. |
| title | Algebra in superextensions of groups, I: zeros and commutativity |
| title_full | Algebra in superextensions of groups, I: zeros and commutativity |
| title_fullStr | Algebra in superextensions of groups, I: zeros and commutativity |
| title_full_unstemmed | Algebra in superextensions of groups, I: zeros and commutativity |
| title_short | Algebra in superextensions of groups, I: zeros and commutativity |
| title_sort | algebra in superextensions of groups, i: zeros and commutativity |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/153373 |
| work_keys_str_mv | AT banakhtt algebrainsuperextensionsofgroupsizerosandcommutativity AT gavrylkivv algebrainsuperextensionsofgroupsizerosandcommutativity AT nykyforchyno algebrainsuperextensionsofgroupsizerosandcommutativity |