Abstract
In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ3 ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x 3) ∨ x ≈ ⊤, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms.
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Both authors are partially supported by Grants MTM2004-031012001-3329 and TIN 2004-07933-C03-02 of Spain and by Grant 2001SGR-0017 of D.G.R. of Generalitat de Catalunya
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Gispert, J., Torrens, A. Axiomatic Extensions of IMT3 Logic. Stud Logica 81, 311–324 (2005). https://doi.org/10.1007/s11225-005-4647-7
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DOI: https://doi.org/10.1007/s11225-005-4647-7