In  a characterization of algebraizability of a sentential logic is obtained in terms of the existence of an isomorphism between the lattice of theories of and the lattice of theories of the equational consequence of its equivalent algebraic semantics , commuting with substitutions. In  we find a generalization of this result in Teorem V.3.5, stating that two deductive systems are equivalent if, and only if, there exists an isomorphism between their lattices of theories, commuting with substitutions. In turn, in  a new generalization of this result is exhibited for Gentzen systems: Theorem 2.19.
Each of these systems is a generalization of the previous one: sentential logics and equational consequences of a class of algebras are particular cases of -deductive systems, which are particular cases of Gentzen systems. But there are other kinds of deductive systems with a similar Isomorphism Theorem that are not, nor extend, Gentzen systems.
A generalization of all these systems are -institutions, which were introduced for the first time by Fiadeiro and Sernadas in , inspired by the work on institutions of Goguen and Burstall in . Institutions cover all these deductive systems and also formalize other multi-sorted ones coming form computing science. -institutions in turn focus attention in the syntax instead of in semantics, as do institutions. For both, a categorical context organizing the information is required.
In  Voutsadakis proved the following result:
Theorem (Voutsadakis). If and are two term -institutions, then they are deductively equivalent if, and only if, there exists an adjoint equivalence that commutes with substitutions.
This result is a generalization of Theorem V.3.5 of . However, in spite of its abstraction level, it is not a generalization of Theorem 2.19 of , since not all Gentzen systems can be exhibited (in fact, only those that are -systems can) as term -institutions. This shows that this condition is probably too strong. But it cannot be just removed: I will provide two -institutions which are not deductively equivalent but with isomorphic categories of theories (through an isomorphism commuting with substitutions).
The objective of eliminating absolutely the conditions over the -institutions in the characterization theorem of deductive equivalence is then vane. However, I will offer a way of extending the result which will cover Gentzen systems among others. To do this, the notion of Grothendieck construction of a -valued functor will be used. We obtain then the following extended version of the Isomorphism Theorem:
Theorem. If and are two
-institutions, then they are deductively equivalent if, and only
if, there exists an adjoint equivalence
that commutes with
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