Тип публикации: статья из журнала
Год издания: 2020
Идентификатор DOI: 10.26516/1997-7670.2020.32.64
Ключевые слова: endomorphism of the groupoid, endomorphisms, groupoids, magmas, monoids∗ this work is supported by the krasnoyarsk mathematical center and financed by the ministry of science and higher education of the russian federation in the framework of the establishment and development of regional centers for mathematics research and education (agreement no. 075-02-2020-1534/1).
Аннотация: Automorphisms and endomorphisms are actively used in various theoretical studies. In particular, the theoretical interest in the study of automorphisms is due to the possibility of representing elements of a group by automorphisms of a certain algebraic system. For example, in 1946, G. Birkhoff showed that each group is the group of all automorphisms of a certain algebra. In 1958, D. Groot published a work in which it was established that every group is a group of all automorphisms of a certain ring. It was established by M. M. Glukhov and G. V. Timofeenko: every finite group is isomorphic to the automorphism group of a suitable finitely defined quasigroup. In this paper, we study endomorphisms of certain finite groupoids with a generating set of k elements and order k+k2, which are not quasigroups and semigroups for k > 1. A description is given of all endomorphisms of these groupoids as mappings of the support, and some structural properties of the monoid of all endomorphisms are established. It was previously established that every finite group embeds isomorphically into the group of all automorphisms of a certain suitable groupoid of order k + k2 and a generating set of k elements. It is shown that for any finite monoid G and any positive integer k ≥ |G| there will be a groupoid S with a generating set of k elements and order k + k2 such that G is isomorphic to some submonoid of the monoid of all endomorphisms of the groupoid S. © 2020 Irkutsk State University. All rights reserved.
Журнал: Bulletin of Irkutsk State University, Series Mathematics
Выпуск журнала: Vol. 32
Номера страниц: 64-78
ISSN журнала: 19977670
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