FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2009, VOLUME 15, NUMBER 7, PAGES 165-177

**Abelian and Hamiltonian groupoids**

A. A. Stepanova

N. V. Trikashnaya

Abstract

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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 $a$,
$b$, $\$\; \backslash bar\; c\; \$$, $\$\; \backslash bar\; d\; \$$ the implication
$\$\; t\; (a,\; \backslash bar\; c)\; =\; t\; (a,\; \backslash bar\; d)$Þ 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 $a,\; b$Î 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.

Location: http://mech.math.msu.su/~fpm/eng/k09/k097/k09708h.htm

Last modified: April 18, 2010