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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2023. Vol 44

On Anti-endomorphisms of Groupoids

Author(s)
Andrey V. Litavrin1

1Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract
In this paper, we study the problem of element-by-element description of the set of all anti-endomorphisms of an arbitrary groupoid. In particular, the structure of the set of all anti-automorphisms of a groupoid is studied. It turned out that the set of all anti-endomorphisms of an arbitrary groupoid decomposes into a union of pairwise disjoint sets of transformations of a special form. These sets of transformations are referred to in this paper as basic sets of anti-endomorphisms. Each base set of anti-endomorphisms is parametrized by some mapping of the support set of the groupoid into a fixed set of two elements. These mappings are called the bipolar anti-endomorphism type. Since the base sets of anti-endomorphisms of various types have an empty intersection, each antiendomorphism can uniquely be associated with its bipolar type. This assignment leads to a bipolar classification of anti-endomorphisms of an arbitrary groupoid. In this paper, we study the semiheap (3-groupoid of a special form) of all anti-endomorphisms. A subsemiheap of anti-endomorphisms of the first type and a subsemiheap of anti-endomorphisms of the second type are constructed. These monotypic semiheaps can be expressed into empty sets for specific groupoids. A conjecture is made about a subsemiheap of a special type of anti-endomorphisms of mixed type. The main research method in this work is the use of internal left and right translations of the groupoid (left and right translations). Since an arbitrary groupoid is considered, the set of all left translations (similarly to right translations) need not be closed with respect to the composition of transformations of the support set of the groupoid.
About the Authors
Andrey V. Litavrin, Cand. Sci. (Phys.–Math.), Assoc. Prof., Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, anm11@rambler.ru
For citation
Litavrin A. V. On Anti-endomorphisms of Groupoids. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 44, pp. 82–97. https://doi.org/10.26516/1997-7670.2023.44.82
Keywords
groupoid endomorphism, anti-endomorphism, groupoid anti-automorphism, bipolar type of groupoid anti-endomorphism, groupoid, anti-endomorphism semiheap, monotypic anti-endomorphism semiheap
UDC
512.577+512.548.2+512.534.2
MSC
20N02, 20M20
DOI
https://doi.org/10.26516/1997-7670.2023.44.82
References
  1. Ayoubi M., Zeglami D. D’Alembert’s functional equations on monoids with an anti-endomorphism. Results Math, 2020, vol. 75, no. 74. https://doi.org/10.1007/s00025-020-01199-z
  2. Ayoubi M., Zeglami D. A variant of D’Alembert’s. functional equation on semigroups with an anti-endomorphism. Aequationes mathematicae, 2022, vol. 96, pp. 549–565. https://doi.org/10.1007/s00010-021-00836-4
  3. Ayoubi M., Zeglami D. D’Alembert’s 𝜇-matrix functional equation on groups with an anti-endomorphism. Mediterranean Journal of Mathematics volume, 2022, vol. 19, no. 219. https://doi.org/10.1007/s00009-022-02129-9
  4. Balaba I.N., Mikhalev A.V. Anti-isomorphisms of graded endomorphism rings of graded modules close to free ones. J. Math. Sci., 2010, vol. 164, no 2. pp. 168–177. https://doi.org/10.1007/s10958-009-9747-x
  5. Beider K.I., Fong Y., Ke W.-F, Wu W.-R. On semi-endomorphisms of groups. Communications in Algebra, 1999, vol. 27, no. 5, pp. 2193-2205. https://doi.org/10.1080/00927879908826558
  6. Belyavskaya G.B., Tabarov A.Kh. Identities with permutations leading to linearity of quasigroups. Discrete Math. Appl., 2009, vol. 19, no. 2, pp. 173–190. https://doi.org/10.4213/dm1037
  7. Gewirtzman L. Anti-isomorphisms of the endomorphism rings of a class of free module. Math. Ann., 1965, vol. 159, pp. 278–284. https://doi.org/10.1007/BF01362446
  8. Gewirtzman L. Anti-isomorphisms of endomorphism rings of torsion-free module. Math. Z., 1967, vol. 98, pp. 391–400. https://doi.org/10.1007/BF01215251
  9. Gornova M.N., Kukina E.G., Romankov V.A. Cryptographic analysis of the Ushakov–Shpilrain authentication protocol based on the binary problem twisted conjugacy. Applied Discrete Mathematics, 2015, vol. 28, no. 2, pp. 46–53. (in Russian) https://doi.org/10.17223/20710410/28/5
  10. Litavrin A.V. Endomorphisms and anti-endomorphisms of some finite groupoids. Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 2022, vol. 24, no. 1, pp. 76–95. (in Russian) https://doi.org/10.15507/2079-6900.24.202201.76-95
  11. Litavrin A.V. Endomorphisms of some groupoids of order 𝑘 + 𝑘2. The Bulletin of Irkutsk State University. Series Mathematics, 2020. vol. 32, pp. 64–78. https://doi.org/10.26516/1997-7670.2020.32.64
  12. Litavrin A.V. On endomorphisms of the additive monoid of subnets of a two-layer neural network. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 39, pp. 111–126. https://doi.org/10.26516/1997-7670.2022.39.111
  13. Litavrin A.V. On an element-by-element description of the monoid of all endomorphisms of an arbitrary groupoid and one classification of endomorphisms of a groupoid. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1. pp. 143–159. (in Russian) https://doi.org/10.21538/0134-4889-2023-29-1-143-159
  14. Litavrin A.V. Endomorphisms of finite commutative groupoids related with multilayer feedforward neural networks. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 1, pp. 130–145. (in Russian) https://doi.org/10.21538/0134-4889-2021-27-1-130-145
  15. Mikhalev A.V., Shatalova M.A. Automorphisms and anti-automorphisms, semigroups of invertible matrices with non-negative elements. Math. Sat., 1970. vol. 81, no. 4, pp. 600–609. (in Russian)
  16. Vagner V.V. Pseudo-semiheaps and semigroups with a transformation. Izv. universities. Mat., 1973, vol. 131, no. 4, pp. 8–15. (in Russian)

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