Functors of Lindenbaum-Tarski, Schematic Interpretations, and Adjoint Cylinders between Sentential Logics
| Autor: | J. Soliveres Tur, J. Climent Vidal |
|---|---|
| Rok vydání: | 2008 |
| Předmět: |
Pointwise
adjoint cylinder Functor Logic schematic interpretation Functor category 18C20 functors of Lindenbaum-Tarski 18A15 Algebra 03B05 Mathematics::Logic Adjoint representation of a Lie algebra Morphism $\ell$-congruence Computer Science::Logic in Computer Science Mathematics::Category Theory 18A40 Natural transformation Adjoint functors 03B20 Monad (category theory) Mathematics |
| Zdroj: | Notre Dame J. Formal Logic 49, no. 2 (2008), 185-202 |
| ISSN: | 0029-4527 |
| DOI: | 10.1215/00294527-2008-007 |
| Popis: | 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 Godel- 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 adjunc- tions 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 Heyt- ing algebras. Finally, we prove that the functor of regularization from Heyt to Bool has, in addition to its well-known right adjoint, i.e., 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, flnally, that such a string of adjunctions constitutes an adjoint cylinder. |
| Databáze: | OpenAIRE |
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