We prove, by using the concept of schematic interpretation, that the natural embedding from the category ISL, of intuitionistic sentential pretheories and i-congruence classes of morphisms, to the category CSL, of classical sentential pretheories and c-congruence classes of morphisms, has a left adjoint, which is related to the double negation interpretation of Gödel-Gentzen, and a right adjoint, which is related to the Law of Excluded Middle. Moreover, we prove that from the left to the right adjoint there is a pointwise (...) epimorphic natural transformation and that since the two endofunctors at CSL, obtained by adequately composing the aforementioned functors, are naturally isomorphic to the identity functor for CSL, the string of adjunctions constitutes an adjoint cylinder. On the other hand, we show that the operators of Lindenbaum-Tarski of formation of algebras from pretheories can be extended to equivalences of categories from the category CSL, respectively, ISL, to the category Bool, of Boolean algebras, respectively, Heyt, of Heyting algebras. Finally, we prove that the functor of regularization from Heyt to Bool has, in addition to its well-known right adjoint (that is, the canonical embedding of Bool into Heyt) a left adjoint, that from the left to the right adjoint there is a pointwise epimorphic natural transformation, and, finally, that such a string of adjunctions constitutes an adjoint cylinder. (shrink)

After defining, for each many-sorted signature Σ = (S, Σ), the category Ter ( Σ ), of generalized terms for Σ (which is the dual of the Kleisli category for , the monad in Set S determined by the adjunction from Set S to Alg ( Σ ), the category of Σ -algebras), we assign, to a signature morphism d from Σ to Λ , the functor from Ter ( Σ ) to Ter ( Λ ). Once defined the mappings (...) that assign, respectively, to a many-sorted signature the corresponding category of generalized terms and to a signature morphism the functor between the associated categories of generalized terms, we state that both mappings are actually the components of a pseudo-functor Ter from Sig to the 2-category Cat . Next we prove that there is a functor Tr Σ , of realization of generalized terms as term operations, from Alg ( Σ ) × Ter ( Σ ) to Set , that simultaneously formalizes the procedure of realization of generalized terms and its naturalness (by taking into account the variation of the algebras through the homomorphisms between them). We remark that from this fact we will get the invariance of the relation of satisfaction under signature change. Moreover, we prove that, for each signature morphism d from Σ to Λ , there exists a natural isomorphism θ d from the functor to the functor , both from the category Alg ( Λ ) × Ter ( Σ ) to the category Set , where d * is the value at d of the arrow mapping of a contravariant functor Alg from Sig to Cat , that shows the invariant character of the procedure of realization of generalized terms under signature change. Finally, we construct the many-sorted term institution by combining adequately the above components (and, in a derived way, the many-sorted specification institution), but for a strict generalization of the standard notion of institution. (shrink)