Abelian and Hamiltonian varieties of groupoids

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.