On an element-by-element description of the monoid of all endomorphisms of an arbitrary groupoid and one classification of endomorphisms of a groupoid (Q6049899)

In this paper a groupoid means a set equipped with a single binary algebraic operation. The author discusses the problem of describing the monoid of all endomorphisms of a groupoid \(G\). He establishes that this monoid can be divided into disjoint classes of endomorphisms, which are referred to as basic sets. These sets are parameterized by mappings \(\gamma :G\rightarrow \left\{ 1,2\right\}\) called bipolar types. The author introduces a classification system for endomorphisms based on these types. He also explores the relationship between the types of endomorphisms in isomorphic groupoids. Additionally, for each type \(\gamma\), the author constructed a subsemigroup of the monoid of all endomorphisms of \(G\). Although these semigroups can degenerate into empty sets, examples of groupoids are given in which these semigroups are nonempty.