Abstract:
We study the anti-endomorphisms of an arbitrary $n$-groupoid (i.e., an algebraic system with a single operation, which is an $n$-ary operation). We construct bipolar classifications of the anti-endomorphisms of an arbitrary $n$-groupoid with index $j$, where $j$ is an arbitrary natural number less than or equal to $n$. These classifications of anti-endomorphisms generalize the bipolar classification of anti-endomorphisms of an ordinary groupoid (i.e., a 2-groupoid). We establish a relationship between bipolar anti-endomorphism types (in the bipolar classification with index $j$) in a pair of isomorphic $n$-groupoids. We obtain formulas for calculating the bipolar type of an arbitrary anti-endomorphism. Semi-heaps (i.e. 3-groupoids with the skew-associativity condition) of anti-endomorphisms are constructed, which consist of anti-endomorphisms of one mixed bipolar type.
Keywords:groupoid, $n$-groupoid, anti-endomorphism, bipolar classification of anti-endomorphisms, bipolar type of anti-endomorphism.
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