Semi-lattice of varieties of quasigroups with linearity
A \(\sigma\)-parastrophe of a class of quasigroups \(\mathfrak{A}\) is a class \({^{\sigma}\mathfrak{A}}\) of all \(\sigma\)-parastrophes of quasigroups from \(\mathfrak{A}\). A set of all pairwise parastrophic classes is called a parastrophic orbit or a truss. A parastrophically closed semi-lattice...
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| Дата: | 2021 |
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Lugansk National Taras Shevchenko University
2021
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-17482021-07-19T08:39:30Z Semi-lattice of varieties of quasigroups with linearity Sokhatsky, F. M. Krainichuk, H. V. Sydoruk, V. A. quasigroup, parastrophe, identity, parastrophic symmetry, parastrophic orbit, truss, bunch, left, right, middle linearity, alinearity, central, semi-central, semi-linear, semi-alinear, linear, alinear variety Primary 20N05, 20N15, 39B52, 08A05; Secondary 05A15, 05B07 A \(\sigma\)-parastrophe of a class of quasigroups \(\mathfrak{A}\) is a class \({^{\sigma}\mathfrak{A}}\) of all \(\sigma\)-parastrophes of quasigroups from \(\mathfrak{A}\). A set of all pairwise parastrophic classes is called a parastrophic orbit or a truss. A parastrophically closed semi-lattice of classes is a bunch.A linearity bunch is a set of varieties which contains the variety of all left linear quasigroups, the variety of all left alinear quasigroups, all their parastrophes and all their intersections. It contains 14 varieties, which are distributed into six parastrophic orbits. All quasigroups from these varieties are called dilinear. To obtain all varieties from the bunch, concepts of middle linearity and middle alinearity are introduced. A well-known identity or a system of identities which describes a variety from every parastrophic orbit of the bunch is cited. An algorithm for obtaining identities which describe all varieties from the parastrophic orbits is given. Examples of quasigroups distinguishing one variety from the other are presented. Lugansk National Taras Shevchenko University Scientific Ukrainian Mathematical School ``Multiary Invertible Functions'' (SUMS ``MIF'') 2021-07-19 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1748 10.12958/adm1748 Algebra and Discrete Mathematics; Vol 31, No 2 (2021) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1748/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1748/807 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1748/811 Copyright (c) 2021 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
quasigroup parastrophe identity parastrophic symmetry parastrophic orbit truss bunch left right middle linearity alinearity central semi-central semi-linear semi-alinear linear alinear variety Primary 20N05 20N15 39B52 08A05; Secondary 05A15 05B07 |
| spellingShingle |
quasigroup parastrophe identity parastrophic symmetry parastrophic orbit truss bunch left right middle linearity alinearity central semi-central semi-linear semi-alinear linear alinear variety Primary 20N05 20N15 39B52 08A05; Secondary 05A15 05B07 Sokhatsky, F. M. Krainichuk, H. V. Sydoruk, V. A. Semi-lattice of varieties of quasigroups with linearity |
| topic_facet |
quasigroup parastrophe identity parastrophic symmetry parastrophic orbit truss bunch left right middle linearity alinearity central semi-central semi-linear semi-alinear linear alinear variety Primary 20N05 20N15 39B52 08A05; Secondary 05A15 05B07 |
| format |
Article |
| author |
Sokhatsky, F. M. Krainichuk, H. V. Sydoruk, V. A. |
| author_facet |
Sokhatsky, F. M. Krainichuk, H. V. Sydoruk, V. A. |
| author_sort |
Sokhatsky, F. M. |
| title |
Semi-lattice of varieties of quasigroups with linearity |
| title_short |
Semi-lattice of varieties of quasigroups with linearity |
| title_full |
Semi-lattice of varieties of quasigroups with linearity |
| title_fullStr |
Semi-lattice of varieties of quasigroups with linearity |
| title_full_unstemmed |
Semi-lattice of varieties of quasigroups with linearity |
| title_sort |
semi-lattice of varieties of quasigroups with linearity |
| description |
A \(\sigma\)-parastrophe of a class of quasigroups \(\mathfrak{A}\) is a class \({^{\sigma}\mathfrak{A}}\) of all \(\sigma\)-parastrophes of quasigroups from \(\mathfrak{A}\). A set of all pairwise parastrophic classes is called a parastrophic orbit or a truss. A parastrophically closed semi-lattice of classes is a bunch.A linearity bunch is a set of varieties which contains the variety of all left linear quasigroups, the variety of all left alinear quasigroups, all their parastrophes and all their intersections. It contains 14 varieties, which are distributed into six parastrophic orbits. All quasigroups from these varieties are called dilinear. To obtain all varieties from the bunch, concepts of middle linearity and middle alinearity are introduced. A well-known identity or a system of identities which describes a variety from every parastrophic orbit of the bunch is cited. An algorithm for obtaining identities which describe all varieties from the parastrophic orbits is given. Examples of quasigroups distinguishing one variety from the other are presented. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2021 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1748 |
| work_keys_str_mv |
AT sokhatskyfm semilatticeofvarietiesofquasigroupswithlinearity AT krainichukhv semilatticeofvarietiesofquasigroupswithlinearity AT sydorukva semilatticeofvarietiesofquasigroupswithlinearity |
| first_indexed |
2024-04-12T06:25:13Z |
| last_indexed |
2024-04-12T06:25:13Z |
| _version_ |
1796109217812185088 |