On retract varieties of algebras

A retract variety is defined as a class of algebras closed under isomorphisms, retracts and products. Let a principal retract variety be generated by one algebra and a set-principal retract variety be generated by some set of algebras. It is shown that (a) not each set-principal retract variety is principal, and (b) not each retract variety is set-principal. A class of connected monounary algebras $\mathcal{S}$ such that every retract variety of monounary algebras is generated by algebras that have all connected components from $\mathcal{S}$ and at most two connected components are isomorphic is defined, this generating class is constructively described. All set-principal retract varieties of monounary algebras are characterized via degree function of monounary algebras.