Abelian and Hamiltonian varieties of groupoids

Abstract

In the work we investigate some groupoids which are the Abelian algebras and the Hamiltonian algebras. An algebra is Abelian if for every polynomial operation and for all elements $a,b,\bar c,\bar d$ the implication $t(a,\bar c)=t(a,\bar d)\rightarrow t(b,\bar c)=t(b,\bar d)$ holds; an algebra is Hamiltonian if every subalgebra is a block of some congruence on the algebra. R.V. 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 Abelian of this algebras.