Integral Classification of Endomorphisms of an Arbitrary Algebra with Finitary Operations

We introduce a bipolar classification with index j for endomorphisms of an arbitrary n-groupoid with n > 1, where j = 1, 2, . . . , n. The classifications of endomorphisms constructed generalize the bipolar classification of endomorphisms of an arbitrary groupoid (i.e., a 2-groupoid) introduced previously. Using a left bipolar classification of endomorphisms of an n-groupoid (a particular case of the obtained classifications), we succeed in constructing an integral classification of endomorphisms of an arbitrary algebra (i.e., a structure without relations) with finitary operations.