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A 2-categorical framework for the syntax and semantics of many-sorted equational logic

Vidal C., Soliveres Tur J.

Reports on Mathematical Logic

2010

Vol. 45

37-95

For, not necessarily similar, single-sorted algebras Fujiwara defined, through the concept of family of basic mappingformulas between single-sorted signatures, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras. Subsequently he also defined an equivalence relation, the relation of conjugation, on the families of basic mapping-formulas. In this article we extend the theory of Fujiwara to the, not necessarily similar, many-sorted algebras, by defining the concept of polyderivor between many-sorted signatures under which are subsumed the standard signature morphisms, the derivors of Goguen- Thatcher-Wagner, and the basic mapping-formulas of Fujiwara.

On the completeness theorem of many-sorted equational logic and the equivalence between Hall algebras and Binabou theories

Vidal J., Tur J.

Reports on Mathematical Logic

2006

Nr 40

127-158

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 Bienabou algebras for a set of sorts S we prove that the category of Bienabou algebras for S is equivalent to the category of Hall algebras for S and isomorphic to the category of Bienabou theories for S, i.e., the many-sorted counterpart of the category of Lawvere theories, hence that Hall algebras and Bienabou theories are equivalent.