Сибирский федеральный университет

Тип публикации: статья из журнала

Год издания: 2023

Идентификатор DOI: 10.1134/s008154382303015x

Ключевые слова: groupoid endomorphism, groupoid automorphism, groupoid, basic set of endomorphisms, bipolar classification of groupoid endomorphisms, monotypic endomorphism semigroups, mathematics, general

Аннотация: The problem of element-by-element description of the monoid of all endomorphisms of an arbitrary groupoid is considered. It is established that this monoid is decomposed into a union of pairwise disjoint classes of endomorphisms; these classes are called basic sets of endomorphisms. Such sets of endomorphisms of a groupoid  $$G$$ are parameterized by mappings $$\gamma:G\to\{1,2\}$$ , which in this paper are called bipolar types (hereinafter, simply types). If some endomorphism belongs to a basic set of type  $$\gamma$$ , then we say that it has type  $$\gamma$$ . Thus, we obtain a classification of all endomorphisms of a fixed groupoid (a bipolar classification of endomorphisms). A connection between the types of endomorphisms of two isomorphic groupoids is revealed. The basic set of endomorphisms need not be closed under composition. Groupoids are constructed in which some basic sets are closed. For each basic set, an endomorphism semigroup contained in this basic set is constructed. These semigroups in some cases degenerate into empty sets. Examples of groupoids are given in which the constructed endomorphism semigroups are nonempty. The constructed semigroups can be used to study the problem of element-by-element description of the monoid of all endomorphisms and to study the structure of the monoid of all endomorphisms.

Журнал: Proceedings of the Steklov Institute of Mathematics

Выпуск журнала: Т.321, №S1

Номера страниц: 170-185

ISSN журнала: 00815438

Место издания: Москва

Издатель: Pleiades Publishing, Ltd. (Плеадес Паблишинг, Лтд)

• Litavrin A. V. (Siberian Federal University)

• Ядро РИНЦ (eLIBRARY.RU)