Conditional rewrite rules as an algebraic semantics of processes

We show that processes defined by equational identities together with conditional rewrite rules in Plotkin style can be given a purely algebraic description. A category of algebraic structures allowing for the specific nature of those languages is defined and its categorical properties are investigated. In particular it is shown how to derive new models from old by adding constraints either on the algebraic structures or on the categories in which their algebras lives.