Pentagonal quasigroups, their translatability and parastrophes
Any pentagonal quasigroup QQ is proved to have the product xy=φ(x)+y−φ(y)xy=\varphi \left(x)+y-\varphi (y), where (Q,+)\left(Q,+) is an Abelian group, φ\varphi is its regular automorphism satisfying φ4−φ3+φ2−φ+ε=0{\varphi }^{4}-{\varphi }^{3}+{\varphi }^{2}-\varphi +\varepsilon =0 and ε\varepsilon...
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De Gruyter
2021
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oai:doaj.org-article:163a26228c004017b35408ae550bf0332021-12-05T14:10:52ZPentagonal quasigroups, their translatability and parastrophes2391-545510.1515/math-2021-0004https://doaj.org/article/163a26228c004017b35408ae550bf0332021-05-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0004https://doaj.org/toc/2391-5455Any pentagonal quasigroup QQ is proved to have the product xy=φ(x)+y−φ(y)xy=\varphi \left(x)+y-\varphi (y), where (Q,+)\left(Q,+) is an Abelian group, φ\varphi is its regular automorphism satisfying φ4−φ3+φ2−φ+ε=0{\varphi }^{4}-{\varphi }^{3}+{\varphi }^{2}-\varphi +\varepsilon =0 and ε\varepsilon is the identity mapping. All Abelian groups of order n<100n\lt 100 inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity (xy⋅x)y⋅x=y\left(xy\cdot x)y\cdot x=y is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is {11n:n=0,1,2,…}\left\{1{1}^{n}:n=0,1,2,\ldots \right\}. We prove that the only translatable commutative pentagonal quasigroup is xy=(6x+6y)(mod11)xy=\left(6x+6y)\left({\rm{mod}}\hspace{0.33em}11). The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the group Zn{{\mathbb{Z}}}_{n} and its automorphism φ(x)=ax\varphi \left(x)=ax is proved to determine the value of aa and the range of values of nn.Dudek Wieslaw A.Monzo Robert A. R.De Gruyterarticlequasigrouppentagonal quasigrouptranslatabilityidempotent20n0220n05MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 184-197 (2021) |
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quasigroup pentagonal quasigroup translatability idempotent 20n02 20n05 Mathematics QA1-939 |
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quasigroup pentagonal quasigroup translatability idempotent 20n02 20n05 Mathematics QA1-939 Dudek Wieslaw A. Monzo Robert A. R. Pentagonal quasigroups, their translatability and parastrophes |
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Any pentagonal quasigroup QQ is proved to have the product xy=φ(x)+y−φ(y)xy=\varphi \left(x)+y-\varphi (y), where (Q,+)\left(Q,+) is an Abelian group, φ\varphi is its regular automorphism satisfying φ4−φ3+φ2−φ+ε=0{\varphi }^{4}-{\varphi }^{3}+{\varphi }^{2}-\varphi +\varepsilon =0 and ε\varepsilon is the identity mapping. All Abelian groups of order n<100n\lt 100 inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity (xy⋅x)y⋅x=y\left(xy\cdot x)y\cdot x=y is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is {11n:n=0,1,2,…}\left\{1{1}^{n}:n=0,1,2,\ldots \right\}. We prove that the only translatable commutative pentagonal quasigroup is xy=(6x+6y)(mod11)xy=\left(6x+6y)\left({\rm{mod}}\hspace{0.33em}11). The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the group Zn{{\mathbb{Z}}}_{n} and its automorphism φ(x)=ax\varphi \left(x)=ax is proved to determine the value of aa and the range of values of nn. |
| format |
article |
| author |
Dudek Wieslaw A. Monzo Robert A. R. |
| author_facet |
Dudek Wieslaw A. Monzo Robert A. R. |
| author_sort |
Dudek Wieslaw A. |
| title |
Pentagonal quasigroups, their translatability and parastrophes |
| title_short |
Pentagonal quasigroups, their translatability and parastrophes |
| title_full |
Pentagonal quasigroups, their translatability and parastrophes |
| title_fullStr |
Pentagonal quasigroups, their translatability and parastrophes |
| title_full_unstemmed |
Pentagonal quasigroups, their translatability and parastrophes |
| title_sort |
pentagonal quasigroups, their translatability and parastrophes |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/163a26228c004017b35408ae550bf033 |
| work_keys_str_mv |
AT dudekwieslawa pentagonalquasigroupstheirtranslatabilityandparastrophes AT monzorobertar pentagonalquasigroupstheirtranslatabilityandparastrophes |
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