Correspondence Theory on Vector Spaces

This paper extends correspondence theory to the framework of \(\mathbb {K}\) -algebras, i.e. vector spaces endowed with a bilinear operation, seen as ‘Kripke frames’. For every \(\mathbb {K}\) -algebra, the lattice of its subspaces can be endowed with the structure of a complete (non necessarily monoidal) residuated lattice. Hence, a sequent of the logic of residuated lattices can be interpreted as a property of its lattice of subspaces. Thus, correspondence theory can be developed between the propositional language of this logic and the first order language of \(\mathbb {K}\) -algebras, analogously to the well known correspondence theory between classical normal modal logic and the first-order language of Kripke frames. In this paper, we develop such a theory for the class of analytic inductive inequalities.