Relational sets and categorical equivalence of algebras

We study relation varieties, i.e. classes of relational sets (resets) of the same type that are closed under the formation of products and retracts. The notions of an irreducible reset and a representation of a reset are defined similarly to the ones for partially ordered sets. We give a characterization of finite irreducible resets. We show that every finite reset has a representation by minimal resets which are certain distinguished irreducible retracts. It turns out that a representation by minimal resets is a smallest one in some sense among all representations of a reset. We prove that non-isomorphic finite irreducible resets generate different relation varieties. We characterize categorical equivalence of algebras via product and retract of certain resets associated with the algebras. In the finite case the characterization involves minimal resets. Examples are given to demonstrate how the general theorems work for particular algebras and resets.