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
Received September 7, 2025
Revised October 16, 2025
Accepted October 20, 2025
Funding Agency: This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation (Agreement no. 075-02-2025-1790).
Andrey Viktorovich Litavrin, Cand. Sci. (Phys.-Math.), Institute of Mathematics and Fundamental Informatics of Sib. Federal. Univer., Krasnoyarsk, 660041 Russia, e-mail: anm11@rambler.ru
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Cite this article as: A.V. Litavrin. On anti-endomorphisms of $n$-groupoids. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 4, pp. 230–246.
