Equivalential unversal Horn theories

As a peliminary point, we start from extending Mal'cev's concept of rational equivalence of prevarieties of pure algebras to those of algebraic systems justifying this extension by a Mal'cev-style categorical characterization. Next, we extend the concept of equivalent purely-algebraic semantics, being a prevariety of pure algebras, to prevarieities of algebraic systems. In this way, the concept of equivalential UHT arises as the respective extension of the one of algebraizable UHT, each equivalential UHT having a unique (modulo rational equivalence) equivalent algebraic semantics. We then apply our general theory of equivalence of universal Horn theories to reducing the problem of finding extensions of an equivalential UHT to that of finding subprevarieties of its equivalent algebraic semantics. Our general elaboration is well-applicable to equivalential sequent calculi with structural rules associated with finitely-valued logics with equality determinant. Finally, we exemplify our general study by exploring four examples of non-algebraizable sequent calculi of such a kind, one of them being equivalent to the corresponding sentential logic.