On Alternating Semigroups of Endomorphisms of a Groupoid

We study the bipolar type of the composition for pairs of endomorphisms of a groupoidand introduce the notion of an alternating pair of endomorphisms. For such a pair, the bipolartype of the composition is represented in terms of the bipolar types of the initial endomorphisms.We suggest an explicit formula for this representation. We also introduce alternating and specialalternating semigroups of endomorphisms of a groupoid so that every pair of endomorphisms froman alternating semigroup is alternating. For every groupoid, we prove that the base set ofendomorphisms of the first type is a special alternating semigroup with identity (i.e., a monoid).For isomorphic groupoids \(G\) and \(G^{\prime }\) , we prove that every special alternating semigroupof endomorphisms of \(G\) is isomorphic toa suitable special alternating semigroup of endomorphisms of \(G^{\prime }\) .