Equivalent universal Horn theories

The aim of this monograph is to introduce and study a general concept of equivalence between universal Horn theories that would cover on an equal formal basis both Mal'cev's concept of rational equivalence between implicational classes of abstract algebras and our more recent concept of definitional equivalence between finitary universal Horn theories without equality proposed and explored in one of our previous works. Starting from a formal definition quite naturally generalizing our conception of definitional equivalence, we then advance our generalized notion towards a Mal'cev-style categorical characterization of equivalence justifying our generalization. We also apply our general study to the issues of equivalential and algebraizable universal Horn theories. We also supply our general results with appropriate counterexamples showing that the former can not be strengthened. At last, we provide an example of a finite finitary infinitely algebraizable universal Horn theory thatjustifies involving infinitary logic within the context of General Algebraic Logic.