Abelian and Hamiltonian varieties of groupoids

We study certain groupoids generating Abelian, strongly Abelian, and Hamiltonian varieties. An algebra is Abelian if t( a,`(c)] ) = t( a,`(d)] ) ? t( b,`(c)] ) = t( b,`(d)] ) t\left( {a,\bar{c}} \right) = t\left( {a,\bar{d}} \right) \to t\left( {b,\bar{c}} \right) = t\left( {b,\bar{d}} \right) for any polynomial operation on the algebra and for all elements a, b, `(c)] \bar{c} , `(d)] \bar{d} . An algebra is strongly Abelian if t( a,`(c)] ) = t( b,`(d)] ) ? t( e,`(c)] ) = t( e,`(d)] ) t\left( {a,\bar{c}} \right) = t\left( {b,\bar{d}} \right) \to t\left( {e,\bar{c}} \right) = t\left( {e,\bar{d}} \right) for any polynomial operation on the algebra and for arbitrary elements a, b, e, `(c)] \bar{c} , `(d)] \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.