Atlas: A deductive system which has associated three different classes of algebras by Felix Bou

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Lattices, Universal Algebra and Applications
May 28-30, 2003
Centro de Algebra da Universidade de Lisboa
Lisbon, Portugal

Organizers
Gabriela Bordalo, Isabel Ferreirim, Maria Joao Saramago, Luis Sequeira

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A deductive system which has associated three different classes of algebras
by
Felix Bou
University of Barcelona (UB)

In Abstract Algebraic Logic a deductive system S is understood as a consequence relation between sets of formulas and formulas satisfying invariance under substitutions. In the literature (see [3]) one can find (at least) the following three general processes for associating a class of algebras with a deductive system:

The variety K_S generated by the Lindenbaum-Tarski algebra (this algebra is the quotient of the generalized matrix of formulas <Fm, Th S > by its Tarski congruence).

The class Alg^* S of algebraic reducts of reduced (according to the Leibniz congruence) matrices for S, which is obtained according to the classical theory of matrices.

The class Alg S of algebraic reducts of reduced (according to the Tarski congruence) generalized matrices of S.

The class Alg^* S has been traditionally regarded as the algebraic counterpart of S; however, there are reasons for considering that it is Alg S (see [3]). It is well known that i) for every S it holds that Alg^* S is a subset of Alg S, which is a subset of K_S, and that ii) Alg^* S = AlgS when S is protoalgebraic. But it has remained as an open question if there is a deductive system such that the three classes of algebras are all different.

We present an affirmative answer exhibiting a deductive system where the three classes of algebras are different. Our result is the following one. Suppose we take S as the strict implication fragment of the global modal consequence associated with the class of all Kripke frames (this is the deductive system sK_\sigma studied in [1]). Then, i) the class K_S is the variety of weakly Heyting algebras introduced in [2], ii) neither Alg^* S nor AlgS are closed under subalgebras, iii) the three associated classes of algebras are diferent.

References

[1] S. Celani and R. Jansana. A closer look at some subintuitionistic logics. Notre Dame Journal of Formal Logic, 200?, to appear.

[2] S. Celani and R. Jansana. Bounded distributive lattices with strict implication. Manuscript, 2000.

[3] J. M. Font and R. Jansana. A general algebraic semantics for sentential logics, volume 7 of Lecture Notes in Logic. Springer-Verlag, 1996.

Date received: February 28, 2003