Relational models for the Lambek calculus with local contraction subexponentials

Abstract The Lambek calculus has a natural class of interpretations on the algebra of binary relations, called R-models. We study extensions of the Lambek calculus with subexponential modalities, which allow structural rules of the local contraction (contracting neighbouring formulae decorated by the modality) and multi-contraction (contracting neighbouring sequences of such formulae). Local contraction has a natural meaning in R-models, namely, it corresponds to the property of a relation to be dense. We consider several logics with the local contraction subexponential, define the corresponding classes of R-models and prove completeness and incompleteness results.