Resumen de The Completeness Theorem for Monads in Categories of Sorted Sets
Juan Carlos Soliveres Tur, Joan B. Climent Vidal 
Birkhoff's completeness theorem of equational logic asserts the coincidence of the model-theoretic and proof-theoretic consequence relations. Goguen and Meseguer, giving a sound and adequate system of inference rules for finitary many-sorted equational deduction, generalized the completeness theorem of Birkhoff to the completeness of finitary many-sorted equational logic and provided simultaneously a full algebraization of finitary many-sorted equational deduction. In this paper, after defining the concepts of equational class and equational theory for a monad in a category of sorted sets and the concept of projective limit-compatible congruence on a category, we prove that the lattice of product-compatible congruences on the dual of the Kleisli category of a monad in a category of sorted sets is identical to the lattice of equational theories for the same monad. In this way we obtain a completeness theorem for monads in categories of sorted sets, and therefore independent of any explicit syntactical representation of the relevant concepts, that generalizes the completeness theorem of Goguen-Meseguer and provides a full categorization of many-sorted equational deduction.
