Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


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Litavrin A. V. Subsystems and Automorphisms of Some Finite Magmas of Order k + k2. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 4, pp. 457-467. DOI: 10.18500/1816-9791-2020-20-4-457-467, EDN: BVWMTL

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Published online: 
30.11.2020
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Russian
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BVWMTL

Subsystems and Automorphisms of Some Finite Magmas of Order k + k2

Autors: 
Litavrin Andrey Viktorovich, Siberian Federal University
Abstract: 

This work is devoted to the study of subsystems of some finite magmas S = (V, ∗) with a generating set of k elements and order k + k2. For k > 1, the magmas S are not semigroups and quasigroups. An element-by-element description of all magmas S subsystems is given. It was found that all the magmas S have subsystems that are semigroups. For k > 1, subsystems that are idempotent nonunit semigroups are explicitly indicated. Previously, a description of an automorphism group was obtained for magmas S. In particular, every symmetric permutation group Sk is isomorphic to the group of all automorphisms of a suitable magma S. In this paper, a general form of automorphism for a wider class of finite magmas of order k + k2 is obtained.

References: 
  1. Bourbaki N. Elements de Mathematique Algebre Chapitres 1 a 3. Springer Science Business Media, 2007. 636 p.
  2. Litavrin A. V. Automorphisms of some magmas of order k + k2. The Bulletin of the Irkutsk State University. Ser. Mathematics, 2018, vol. 26, pp. 47–61 (in Russian). DOI: https://doi.org/10.26516/1997-7670.2018.26.47
  3. Maltsev A. I. On the multiplication of classes of algebraic systems. Sib. Mat. Jour. [Siberian Mathematical Journal], 1967, vol. 8, no. 2, pp. 346–365 (in Russian).
  4. Knyazev O. V. On the groupoid of varieties of completely simple semigroups. Soviet Math. (Iz. VUZ), 1988, vol. 32, no. 10, pp. 1–12.
  5. Martynova T. A. On the product of semigroup varieties. Soviet Math. (Iz. VUZ), 1988, vol. 32, no. 1, pp. 43–50.
  6. Martynov L. M. On the multiplication of varieties of algebras. Russian Math. (Iz. VUZ), 1994, vol. 38, no. 11, pp. 50–55.
  7. Jones P. R. Mal’cev products of varieties of completely regular semigroups. J. Austral. Math. Soc., 1987, vol. 42, iss. 2, pp. 227–246. DOI: https://doi.org/10.1017/S1446788700028226
  8. Day A. Idempotents in the groupoid of all SP classes of lattices. Canad. Math. Bull., 1978, vol. 21, iss. 4, pp. 499–501. DOI: https://doi.org/10.4153/CMB-1978-085-2
  9. Gr¨atzer G., Kelly D. Products of lattice varieties. Algebra Universalis, 1985, vol. 21, iss. 1, pp. 33–45. DOI: https://doi.org/10.1007/BF01187554
  10. Novikov B. V. On decomposition of Moufang groupoids. Quasigroups Related Systems, 2008, vol. 16. no. 1, pp. 97–101.
  11. Belyavskaya G. B., Tabarov A. Kh. Groupoids with the identity defining the commutative Moufang loops. J. Math. Sci., 2010, vol. 164, iss. 1, pp. 21–25. DOI: https://doi.org/10.1007/s10958-009-9733-3
  12. Shcherbakov V. A., Tabarov A. Kh., Puskash D. I. On congruences of groupoids closely connected with quasigroups. J. Math. Sci., 2009, vol. 163, iss. 6, pp. 785–795. DOI: https://doi.org/10.1007/s10958-009-9716-4
  13. Stepanova A. A., Trikashnaya N. V. Abelian and Hamiltonian groupoids. J. Math. Sci., 2010, vol. 169, iss. 5, pp. 671–679. DOI: https://doi.org/10.1007/s10958-010-0068-x
  14. Gluskin L. M. Positional operatives. Mat. sb. [Sbornik: Mathematics], 1965, vol. 68 (110), iss. 3, pp. 444–472 (in Russian).
  15. Davidov S. S. On the Structure of Medial Divisible n-Ary Groupoids. Math. Notes, 2018, vol. 104, no. 1, pp. 29–38. DOI: https://doi.org/10.1134/S0001434618070040
  16. Davidov S. S. Free commutative medial n-ary groupoids. Discrete Math. Appl., 2015, vol. 25, iss. 4, pp. 203–210. DOI: https://doi.org/10.1515/dma-2015-0020
  17. Davidov S. S. On the solvability of the equational theory of commutative medial n-ary groupoids. Discrete Math. Appl., 2013, vol. 23, iss. 2, pp. 125–143. DOI: https://doi.org/10.1515/dma-2013-007
  18. Gal’mak A. M. On non-n-semiabelianism polyadic groupoids of special class. PFMT [Problems of Physics, Mathematics and Technology], 2019, vol. 38, no. 1, pp. 31–39 (in Russian).
  19. Gal’mak A. M. Permutability of elements in polyadic groupoids of special form. PFMT [Problems of Physics, Mathematics and Technology], 2018, vol. 36, no. 3, pp. 70–75 (in Russian).
  20. Nazarov M. N. A Self-Induced Metric on Groupoids and its Application to the Analysis of Cellular Interactions in Biology. J. Math. Sci., 2015, vol. 206, iss. 5, pp. 561–569. DOI: https://doi.org/10.1007/s10958-015-2333-5
  21. Katyshev S. Yu., Markov V. T., Nechaev A. A. Application of non-associative groupoids to the realization of an open key distribution procedure. Discrete Math. Appl., 2015, vol. 25, iss. 1, pp. 9–24. DOI: https://doi.org/10.1515/dma-2015-0002
  22. Baryshnikov A. V., Katyshev S. Yu. Application of non-associative structures to the construction of public key distribution algorithms. Matematicheskiye voprosy kriptografii [Mathematical Aspects of Cryptography], 2018, vol. 9, iss. 4, pp. 5–30 (in Russian).
  23. Markov V. T., Mikhalev A. V., Nechaev A. A. Nonassociative algebraic structures in cryptography and coding. Fundam. Prikl. Mat., 2016, vol. 21, iss. 4, pp. 99–124 (in Russian).
  24. Bredikhin D. A. On Elasses of Groupoids of Relations with Diophantine Operations. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2013, vol. 13, iss. 4, pt. 2, pp. 28–34 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2013-13-4-28-34
  25. Bredikhin D. A. Identities of Groupoids of Relations With Operation of Cylindered Intersection. Russian Math. (Iz. VUZ), 2018, vol. 62, no. 8, pp. 9–16. DOI: https://doi.org/10.3103/S1066369X18080029
  26. Bredikhin D. A. On bases of identities for varieties of groupoids of relations. Chebyshevskii Sbornik, 2018, vol. 19, no. 1, pp. 26–34 (in Russian). DOI: https://doi.org/10.22405/2226-8383-2018-19-1-26-34
  27. Gluskin L. M. Automorphisms of semigroups of binary relations. Matem. zap. Ural. gos. un-ta [Mathematical notes of the Ural State University], 1967, vol. 6, pp. 44–54 (in Russian).
  28. Gluskin L. M. Automorphisms of multiplicative semigroups of matrix algebras. Uspekhi Mat. Nauk, 1956, vol. 11, iss. 1 (67), pp. 199–206 (in Russian).
  29. Halezov E. A. Automorphisms of matrix semigroups. Doklady Akademii nauk SSSR [Soviet Mathematics Doklady], 1954, vol. 96, no. 2, pp. 245–248 (in Russian).
  30. Bunina E. I., Semenov P. P. Automorphisms of the semigroup of invertible matrices with nonnegative elements over commutative partially ordered rings. Fundam. Prikl. Mat., 2008, vol. 14, iss. 4, pp. 75–85; J. Math. Sci., 2009, vol. 162, iss. 5, pp. 633–655. DOI: https://doi./10.1007/s10958-009-9650-5
  31. Khalezov E. A. Automorphisms of primitive quasigroups. Mat. Sb. [Sbornik: Mathematics], 1961, vol. 53 (95), no. 3, pp. 329–342 (in Russian).
  32. Shmatkov V. D. Isomorphisms and Automorphisms of Matrix Algebras Over Lattices. J. Math. Sci., 2015, vol. 211, iss. 3, pp. 434–440. DOI: https://doi.org/10.1007/s10958-015-2614-z
  33. Il’inykh A. P. Classification of finite groupoids with 2-transitive automorphism groups. Russian Acad. Sci. Sb. Math., 1995, vol. 82, no. 1, pp. 175–197.
  34. Ilinykh A. P. Groupoids of order q(q ± 1)/2, q = 2r with automorphism group isomorphic to SL(2, q). Sib. Math. J., 1995, vol. 36, no. 6, pp. 1159–1163. DOI: https://doi.org/10.1007/BF02106838
  35. Litavrin A. V. Automorphisms of some finite magmas with an order strictly less than the number N(N +1) and a generating set of N elements. Vestnik TVGU. Ser. Prikl. Matem. [Herald of Tver State University. Ser. Applied Mathematics], 2019, iss. 2, pp. 70–87 (in Russian). DOI: https://doi.org/10.26456/vtpmk533
  36. Maltsev A. I. Algebraicheskie sistemy [Algebraic systems]. Moscow, Nauka, 1970. 392 p. (in Russian).
Received: 
01.09.2019
Accepted: 
30.09.2019
Published: 
30.11.2020