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2008 Functors of Lindenbaum-Tarski, Schematic Interpretations, and Adjoint Cylinders between Sentential Logics
J. Soliveres Tur, J. Climent Vidal
Notre Dame J. Formal Logic 49(2): 185-202 (2008). DOI: 10.1215/00294527-2008-007

Abstract

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.

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J. Soliveres Tur. J. Climent Vidal. "Functors of Lindenbaum-Tarski, Schematic Interpretations, and Adjoint Cylinders between Sentential Logics." Notre Dame J. Formal Logic 49 (2) 185 - 202, 2008. https://doi.org/10.1215/00294527-2008-007

Information

Published: 2008
First available in Project Euclid: 15 May 2008

zbMATH: 1155.03048
MathSciNet: MR2402041
Digital Object Identifier: 10.1215/00294527-2008-007

Subjects:
Primary: 03B05 , 03B20
Secondary: 18A15 , 18A40 , 18C20

Keywords: $\ell$-congruence , adjoint cylinder , functors of Lindenbaum-Tarski , schematic interpretation

Rights: Copyright © 2008 University of Notre Dame

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Vol.49 • No. 2 • 2008
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