Geodesic
Abelian and Hamiltonian groupoids
Fundamentalʹnaâ i prikladnaâ matematika, Tome 15 (2009) no. 7, pp. 165-177 
A. A. Stepanova; N. V. Trikashnaya. Abelian and Hamiltonian groupoids. Fundamentalʹnaâ i prikladnaâ matematika, Tome 15 (2009) no. 7, pp. 165-177. http://geodesic.mathdoc.fr/item/FPM_2009_15_7_a7/
@article{FPM_2009_15_7_a7,
     author = {A. A. Stepanova and N. V. Trikashnaya},
     title = {Abelian and {Hamiltonian} groupoids},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {165--177},
     year = {2009},
     volume = {15},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2009_15_7_a7/}
}
TY  - JOUR
AU  - A. A. Stepanova
AU  - N. V. Trikashnaya
TI  - Abelian and Hamiltonian groupoids
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2009
SP  - 165
EP  - 177
VL  - 15
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/FPM_2009_15_7_a7/
LA  - ru
ID  - FPM_2009_15_7_a7
ER  - 
%0 Journal Article
%A A. A. Stepanova
%A N. V. Trikashnaya
%T Abelian and Hamiltonian groupoids
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2009
%P 165-177
%V 15
%N 7
%U http://geodesic.mathdoc.fr/item/FPM_2009_15_7_a7/
%G ru
%F FPM_2009_15_7_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

In this work, we investigate some groupoids that are Abelian algebras and Hamiltonian algebras. An algebra is Abelian if for every polynomial operation and for all elements $a,b,\bar c,\bar d$ the implication $t(a,\bar c)=t(a,\bar d)\Longrightarrow t(b,\bar c)=t(b,\bar d)$ holds. An algebra is Hamiltonian if every subalgebra is a block of some congruence on the algebra. R. J. Warne in 1994 described the structure of the Abelian semigroups. In this work, we describe the Abelian groupoids with identity, the Abelian finite quasigroups, and the Abelian semigroups $S$ such that $abS=aS$ and $Sba=Sa$ for all $a,b\in S$. We prove that a finite Abelian quasigroup is a Hamiltonian algebra. We characterize the Hamiltonian groupoids with identity and semigroups under the condition of Abelianity of these algebras.

[1] Belousov V. D., Osnovy teorii kvazigrupp i lup, Nauka, M., 1967 | MR | Zbl

[2] Klifford A., Preston G., Algebraicheskaya teoriya polugrupp, Mir, M., 1972 | Zbl

[3] Ovchinnikova E. V., “Ob abelevykh gruppoidakh s obrazami maloi moschnosti”, Algebra i teoriya modelei, Izd-vo NGTU, Novosibirsk, 2005

[4] Khobbi D., Makkenzi R., Stroenie konechnykh algebr, Mir, M., 1993 | MR

[5] Kiss E. W., Valeriote M. A., “Strongly Abelian varieties and the Hamiltonian property”, Can. J. Math., 43:2 (1991), 1–16 | DOI | MR

[6] Kiss E. W., Valeriote M. A., “Abelian algebras and the Hamiltonian property”, J. Pure Appl. Algebra, 87:1 (1993), 37–49 | DOI | MR | Zbl

[7] Warne R. J., “Semigroups obeying the term condition”, Algebra Universalis, 31:1 (1994), 113–123 | DOI | MR | Zbl

[8] Warne R. J., “TC semigroups and inflations”, Semigroup Forum., 54:1 (1997), 271–277 | DOI | MR | Zbl