Multi-valued logic and Gröner bases with applications to modal logic

In the case of classical logic, the Stone isomorphism between Boolean Algebras and Boolean Rings is at the basis of the methods which reduce a logical problem to an algebraic one about polynomials. In this paper, we generalize this kind of reduction to the case of any multi-valued logic. Our main result is the Theorem 4.4 which transforms a deduction problem in a multi-valued logic to an equivalent problem about ideal membership in a polynomial ring. We give some examples of applications; for instance we detail the case of Lukasiewicz's modal logic