On the Completeness Theorem of Many-sorted Equational Logic and the Equivalence between Hall Algebras and Bénabou Theories

The completeness theorem of equational logic of Birkhoff 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 many-sorted deduction, founded ultimately on the congruences on Hall algebras, generalized the completeness theorem of Birkhoff to the completeness theorem of many-sorted equational logic. In this paper, after simplifying the specification of Hall algebras as given by Goguen-Meseguer, we obtain another many-sorted equational calculus from which we prove that the inference rules of abstraction and concretion due to Goguen-Meseguer are derived rules. Finally, after defining the Bénabou algebras for a set of sorts S we prove that the category of Bénabou algebras for S is equivalent to the category of Hall algebras for S and isomorphic to the category of Bénabou theories for S, i.e., the many-sorted counterpart of the category of Lawvere theories, hence that Hall algebras and Bénabou theories are equivalent.