Equivalential logics (I)

The class of equivalential logics comprises all implicative logics in the sense of Rasiowa 9], Suszko's logicSCI and many Others. Roughly speaking, a logic is equivalential iff the greatest strict congruences in its matrices (models) are determined by polynomials. The present paper is the first part of the survey in which systematic investigations into this class of logics are undertaken. Using results given in 3] and general theorems from the theory of quasi-varieties of models 5] we give a characterization of all simpleC-matrices for any equivalential logicC (Theorem I.14). In corollaries we give necessary and sufficient conditions for the class of all simple models for a given equivalential logic to be closed under free products (Theorem I.18). Theorem I.17 can be generalized as follows:For any equivalential logic C, clauses (i), (iii)and (v),formulated in Th.I.17,are equivalent.