On the Structure of Set-Theoretic Polygon Equations
Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory associated with oriented simplices. Where...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Author: | |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212256 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On the Structure of Set-Theoretic Polygon Equations. Folkert Müller-Hoissen. SIGMA 20 (2024), 051, 30 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862708040091828224 |
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| author | Müller-Hoissen, Folkert |
| author_facet | Müller-Hoissen, Folkert |
| citation_txt | On the Structure of Set-Theoretic Polygon Equations. Folkert Müller-Hoissen. SIGMA 20 (2024), 051, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory associated with oriented simplices. Whereas the ( − 1)-simplex equation can be regarded as a realization of the higher Bruhat order (, − 2), the -gon equation is a realization of the higher Tamari order (, − 2). The latter and its dual ~(, − 2), associated with which is the dual -gon equation, have been shown to arise as suborders of (N, N−2) via a ''three-color decomposition''. There are two different reductions of (, − 2) and ~(, − 2), to ( − 1, − 3), respectively ~( − 1, − 3). In this work, we explore the reductions of (dual) polygon equations, which yield relations between solutions of neighboring (dual) polygon equations. We also elaborate (dual) polygon equations in this respect explicitly up to the octagon equation.
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| first_indexed | 2026-03-19T05:23:28Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212256 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T05:23:28Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Müller-Hoissen, Folkert 2026-02-03T07:55:32Z 2024 On the Structure of Set-Theoretic Polygon Equations. Folkert Müller-Hoissen. SIGMA 20 (2024), 051, 30 pages 1815-0659 2020 Mathematics Subject Classification: 06A06; 06A07; 15A69; 16T05; 16T25; 17A01; 18D10 arXiv:2305.17974 https://nasplib.isofts.kiev.ua/handle/123456789/212256 https://doi.org/10.3842/SIGMA.2024.051 Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory associated with oriented simplices. Whereas the ( − 1)-simplex equation can be regarded as a realization of the higher Bruhat order (, − 2), the -gon equation is a realization of the higher Tamari order (, − 2). The latter and its dual ~(, − 2), associated with which is the dual -gon equation, have been shown to arise as suborders of (N, N−2) via a ''three-color decomposition''. There are two different reductions of (, − 2) and ~(, − 2), to ( − 1, − 3), respectively ~( − 1, − 3). In this work, we explore the reductions of (dual) polygon equations, which yield relations between solutions of neighboring (dual) polygon equations. We also elaborate (dual) polygon equations in this respect explicitly up to the octagon equation. Some important insights that led to this work originated from my collaboration with Aristophanes Dimakis, who, sadly, passed away in 2021. I would like to thank Jim Stasheff for helpful correspondence. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Structure of Set-Theoretic Polygon Equations Article published earlier |
| spellingShingle | On the Structure of Set-Theoretic Polygon Equations Müller-Hoissen, Folkert |
| title | On the Structure of Set-Theoretic Polygon Equations |
| title_full | On the Structure of Set-Theoretic Polygon Equations |
| title_fullStr | On the Structure of Set-Theoretic Polygon Equations |
| title_full_unstemmed | On the Structure of Set-Theoretic Polygon Equations |
| title_short | On the Structure of Set-Theoretic Polygon Equations |
| title_sort | on the structure of set-theoretic polygon equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212256 |
| work_keys_str_mv | AT mullerhoissenfolkert onthestructureofsettheoreticpolygonequations |