On the bipolar classification of endomorphisms of a groupoid

In this paper, a method is obtained for calculating the bipolar type of endomorphism of an arbitrary groupoid. For groupoids with pairwise distinct left translations of elements, the described method for calculating the bipolar type of an endomorphism leads to a criterion for the fixed point of a given endomorphism. In particular, such groupoids include groupoids with a right neutral element, monoids, loops and groups. It turned out that the bipolar type of endomorphisms of a groupoid with pairwise distinct left translations ones contains all the information about the fixed points of endomorphisms of this type. A basic set of endomorphisms of a group is established, containing all regular automorphisms. A method is found for calculating the bipolar type of an inner automorphism of a monoid. We obtain upper bounds for the order of the monoid of all endomorphisms (and the group of all automorphisms) of an algebraic system with finite support that has a binary algebraic operation.