Gröbner–Shirshov Bases Theory for Trialgebras
We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zh...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/11/1207 |
| _version_ | 1797532552871804928 |
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| author | Juwei Huang Yuqun Chen |
| author_facet | Juwei Huang Yuqun Chen |
| author_sort | Juwei Huang |
| collection | DOAJ |
| description | We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand–Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated, respectively. |
| first_indexed | 2024-03-10T11:00:54Z |
| format | Article |
| id | doaj.art-d012e28349e544ad90aa03d24affa743 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T11:00:54Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d012e28349e544ad90aa03d24affa7432023-11-21T21:31:04ZengMDPI AGMathematics2227-73902021-05-01911120710.3390/math9111207Gröbner–Shirshov Bases Theory for TrialgebrasJuwei Huang0Yuqun Chen1School of Mathematical Sciences, South China Normal University, Guangzhou 510631, ChinaSchool of Mathematical Sciences, South China Normal University, Guangzhou 510631, ChinaWe establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand–Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated, respectively.https://www.mdpi.com/2227-7390/9/11/1207Gröbner–Shirshov basisnormal formGelfand–Kirillov dimensiontrialgebratrisemigroup |
| spellingShingle | Juwei Huang Yuqun Chen Gröbner–Shirshov Bases Theory for Trialgebras Mathematics Gröbner–Shirshov basis normal form Gelfand–Kirillov dimension trialgebra trisemigroup |
| title | Gröbner–Shirshov Bases Theory for Trialgebras |
| title_full | Gröbner–Shirshov Bases Theory for Trialgebras |
| title_fullStr | Gröbner–Shirshov Bases Theory for Trialgebras |
| title_full_unstemmed | Gröbner–Shirshov Bases Theory for Trialgebras |
| title_short | Gröbner–Shirshov Bases Theory for Trialgebras |
| title_sort | grobner shirshov bases theory for trialgebras |
| topic | Gröbner–Shirshov basis normal form Gelfand–Kirillov dimension trialgebra trisemigroup |
| url | https://www.mdpi.com/2227-7390/9/11/1207 |
| work_keys_str_mv | AT juweihuang grobnershirshovbasestheoryfortrialgebras AT yuqunchen grobnershirshovbasestheoryfortrialgebras |