Geodesic
Endomorphisms of some groupoids of order $k+k^2$
The Bulletin of Irkutsk State University. Series Mathematics, Tome 32 (2020), pp. 64-78 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Automorphisms and endomorphisms are actively used in various theoretical studies. In particular, the theoretical interest in the study of automorphisms is due to the possibility of representing elements of a group by automorphisms of a certain algebraic system. For example, in 1946, G. Birkhoff showed that each group is the group of all automorphisms of a certain algebra. In 1958, D. Groot published a work in which it was established that every group is a group of all automorphisms of a certain ring. It was established by M. M. Glukhov and G. V. Timofeenko: every finite group is isomorphic to the automorphism group of a suitable finitely defined quasigroup. In this paper, we study endomorphisms of certain finite groupoids with a generating set of $k$ elements and order $k + k^2$, which are not quasigroups and semigroups for $k>1$. A description is given of all endomorphisms of these groupoids as mappings of the support, and some structural properties of the monoid of all endomorphisms are established. It was previously established that every finite group embeds isomorphically into the group of all automorphisms of a certain suitable groupoid of order $ k + k^2$ and a generating set of $k$ elements. It is shown that for any finite monoid $G$ and any positive integer $k\ge|G| $ there will be a groupoid $S$ with a generating set of $k$ elements and order $k+k^2$ such that $G$ is isomorphic to some submonoid of the monoid of all endomorphisms of the groupoid $S$.
Keywords: endomorphisms, magmas, monoids.
Mots-clés : endomorphism of the groupoid, groupoids 
@article{IIGUM_2020_32_a4,
     author = {A. V. Litavrin},
     title = {Endomorphisms of some groupoids of order $k+k^2$},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {64--78},
     year = {2020},
     volume = {32},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2020_32_a4/}
}
TY  - JOUR
AU  - A. V. Litavrin
TI  - Endomorphisms of some groupoids of order $k+k^2$
JO  - The Bulletin of Irkutsk State University. Series Mathematics
PY  - 2020
SP  - 64
EP  - 78
VL  - 32
UR  - http://geodesic.mathdoc.fr/item/IIGUM_2020_32_a4/
LA  - en
ID  - IIGUM_2020_32_a4
ER  - 
%0 Journal Article
%A A. V. Litavrin
%T Endomorphisms of some groupoids of order $k+k^2$
%J The Bulletin of Irkutsk State University. Series Mathematics
%D 2020
%P 64-78
%V 32
%U http://geodesic.mathdoc.fr/item/IIGUM_2020_32_a4/
%G en
%F IIGUM_2020_32_a4
A. V. Litavrin. Endomorphisms of some groupoids of order $k+k^2$. The Bulletin of Irkutsk State University. Series Mathematics, Tome 32 (2020), pp. 64-78. http://geodesic.mathdoc.fr/item/IIGUM_2020_32_a4/

[1] Bunina E. I., Semenov P. P., “Automorphisms of the semigroup of invertible matrices with nonnegative elements over commutative partially ordered rings”, J. Math. Sci., 162:5 (2009), 633–655 | DOI | MR | MR | Zbl

[2] Birkgof G. O., “On groups of automorphisms”, Revista de la Union Math. Argentina, 1946, no. 4, 155–157

[3] Katyshev S.YU., Markov V. T., Nechayev A. A., “Application of non-associative groupoids to the realization of an open key distribution procedure”, Discrete Mathematics and Applications, 25:1 (2015), 9–24 | DOI | MR | Zbl

[4] Kurosh A. G., Lectures on General Algebra, Lan Publ, St. Petersburg, 2007, 560 pp. (in Russian) | MR

[5] Litavrin A. V., “Automorphisms of some magmas of order $k + k^2 $”, The Bulletin of Irkutsk State University. Series Mathematics, 26 (2018), 47–61 (in Russian) | DOI | MR | Zbl

[6] Litavrin A. V., “Automorphisms of some finite magmas with order strictly less than the number N (N +1) and a generating set of N elements”, Vestnik TvGU. Series Prikladnaya matematika, 2018, no. 2, 70–87 (in Russian) | DOI

[7] Mikhalev A. V., Shatalova M. A., “Automorphisms and anti-automorphisms, semigroups of invertible matrices with nonnegative elements”, Math. sbornik, 81:4 (1970), 600–609 (in Russian) | Zbl

[8] Plotkin B. I., Groups of automorphisms of algebraic systems, Nauka Publ, M., 1966 | MR

[9] Semenov P. P., “Endomorphisms of semigroups of invertible nonnegative matrices over ordered rings”, J. Math. Sci., 193:4 (2013), 591–600 | DOI | MR | Zbl

[10] Tabarov A. Kh., “Homomorphisms and endomorphisms of linear and alinear quasigroups”, Discrete Math. Appl., 17:3 (2007), 253–260 | DOI | MR | MR | Zbl

[11] Timofeyenko G. V., Glukhov M. M., “Automorphism group of finitely defined quasigroups”, Matem. zametki, 37:5 (1985), 617–626 (in Russian) | MR | Zbl

[12] Groot J., “Automorphism groups of rings”, Int. Congr. of Mathematicians (Edinburgh, 1958), 18 pp.