Geodesic
Abelian and Hamiltonian varieties of groupoids
Algebra i logika, Tome 50 (2011) no. 3, pp. 388-398.

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

We study certain groupoids generating Abelian, strongly Abelian, and Hamiltonian varieties. An algebra is Abelian if $t(a,\bar c)=t(a,\bar d)\to t(b,\bar c)=t(b,\bar d)$ for any polynomial operation on the algebra and for all elements $a,b,\bar c,\bar d$. An algebra is strongly Abelian if $t(a,\bar c)=t(b,\bar d)\to t(e,\bar c)=t(e,\bar d)$ for any polynomial operation on the algebra and for arbitrary elements $a,b,e,\bar c,\bar d$. An algebra is Hamiltonian if any subalgebra of the algebra is a congruence class. A variety is Abelian (strongly Abelian, Hamiltonian) if all algebras in a respective class are Abelian (strongly Abelian, Hamiltonian). We describe semigroups, groupoids with unity, and quasigroups generating Abelian, strongly Abelian, and Hamiltonian varieties.
Keywords: Abelian algebra, Hamiltonian algebra, semigroup.
Mots-clés : groupoid, quasigroup 
@article{AL_2011_50_3_a5,
     author = {A. A. Stepanova and N. V. Trikashnaya},
     title = {Abelian and {Hamiltonian} varieties of groupoids},
     journal = {Algebra i logika},
     pages = {388--398},
     publisher = {mathdoc},
     volume = {50},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2011_50_3_a5/}
}
TY  - JOUR
AU  - A. A. Stepanova
AU  - N. V. Trikashnaya
TI  - Abelian and Hamiltonian varieties of groupoids
JO  - Algebra i logika
PY  - 2011
SP  - 388
EP  - 398
VL  - 50
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2011_50_3_a5/
LA  - ru
ID  - AL_2011_50_3_a5
ER  - 
%0 Journal Article
%A A. A. Stepanova
%A N. V. Trikashnaya
%T Abelian and Hamiltonian varieties of groupoids
%J Algebra i logika
%D 2011
%P 388-398
%V 50
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2011_50_3_a5/
%G ru
%F AL_2011_50_3_a5
A. A. Stepanova; N. V. Trikashnaya. Abelian and Hamiltonian varieties of groupoids. Algebra i logika, Tome 50 (2011) no. 3, pp. 388-398. http://geodesic.mathdoc.fr/item/AL_2011_50_3_a5/

[1] D. Khobbi, R. MakKenzi, Stroenie konechnykh algebr, Mir, M., 1993 | MR

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

[3] E. Kiss, M. Valeriote, “Strongly abelian varieties and the Hamiltonian property”, Can. J. Math., 43:2 (1991), 331–346 | DOI | MR | Zbl

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

[5] R. J. Warne, “TC semigroups and inflations”, Semigroup Forum, 54:3 (1997), 271–277 | DOI | MR | Zbl

[6] A. A. Stepanova, N. V. Trikashnaya, “Abelevy i gamiltonovy gruppoidy” (to appear)

[7] E. V. Ovchinnikova, “Ob abelevykh gruppoidakh s obrazami maloi moschnosti”, Algebra i teoriya modelei, NGTU, 2005, 125–131 | MR

[8] A. I. Maltsev, Algebraicheskie sistemy, Nauka, M., 1970 | MR

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