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Algebra Logika, 2011, Volume 50, Number 3, Pages 388–398 (Mi al492)  

Abelian and Hamiltonian varieties of groupoids

A. A. Stepanova, N. V. Trikashnaya

Institute of Mathematics and Computer Sciences, Far Eastern State University, Vladivostok, Russia

Abstract: 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, groupoid, quasigroup, semigroup.

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English version:
Algebra and Logic, 2011, 50:3, 272–278

Bibliographic databases:

UDC: 510.8+512.57
Received: 05.05.2010
Revised: 03.04.2011

Citation: A. A. Stepanova, N. V. Trikashnaya, “Abelian and Hamiltonian varieties of groupoids”, Algebra Logika, 50:3 (2011), 388–398; Algebra and Logic, 50:3 (2011), 272–278

Citation in format AMSBIB
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