Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence
In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their bi...
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MDPI AG
2021-04-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/9/972 |
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| author | Steven Duplij |
| author_facet | Steven Duplij |
| author_sort | Steven Duplij |
| collection | DOAJ |
| description | In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and <i>R</i>-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, <i>n</i>-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case. |
| first_indexed | 2024-03-10T11:56:54Z |
| format | Article |
| id | doaj.art-47a7bc1ead904b0f821df5f8776a01e0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T11:56:54Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-47a7bc1ead904b0f821df5f8776a01e02023-11-21T17:16:15ZengMDPI AGMathematics2227-73902021-04-019997210.3390/math9090972Higher Braid Groups and Regular Semigroups from Polyadic-Binary CorrespondenceSteven Duplij0Center for Information Technology (WWU IT), Universität Münster, Röntgenstrasse 7-13, D-48149 Münster, GermanyIn this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and <i>R</i>-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, <i>n</i>-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.https://www.mdpi.com/2227-7390/9/9/972regular semigroupbraid groupgeneratorrelationpresentationCoxeter group |
| spellingShingle | Steven Duplij Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence Mathematics regular semigroup braid group generator relation presentation Coxeter group |
| title | Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence |
| title_full | Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence |
| title_fullStr | Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence |
| title_full_unstemmed | Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence |
| title_short | Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence |
| title_sort | higher braid groups and regular semigroups from polyadic binary correspondence |
| topic | regular semigroup braid group generator relation presentation Coxeter group |
| url | https://www.mdpi.com/2227-7390/9/9/972 |
| work_keys_str_mv | AT stevenduplij higherbraidgroupsandregularsemigroupsfrompolyadicbinarycorrespondence |