Abstract:
The bipolar types of composition of a pair of endomorphisms of a groupoid are studied in this work. The notion of an alternating pair of endomorphisms of a groupoid is introduced. For such pairs, a formula is established for calculating the bipolar type of a composition using the bipolar types of endomorphisms included in the composition. Alternating and special alternating semigroups of endomorphisms of a groupoid are introduced. Any two endomorphisms from an alternating endomorphism semigroup form an alternating pair. It is shown that the basic set of endomorphisms of the first type is a special alternating semigroup with identity (that is, a monoid). We study the connection between special alternating endomorphism semigroups of two isomorphic groupoids $G$ and $G'$. It is established that every special alternating semigroup of endomorphisms of the groupoid $G$ is isomorphic to some special alternating semigroup of the groupoid $G'$.
Key words:groupoid endomorphism, groupoid, base set of endomorphisms, monotypic endomorphism semigroups, multitype semigroups of endomorphisms, alternating groupoid endomorphism semigroups, special alternating endomorphism semigroups, bipolar type of composition.
Fulltext:
PDF file (390 kB)