Multi-Sorted Residuation

Nonassociative Lambek Calculus (NL) is a pure logic of residuation, involving one binary operation (product) and its two residual operations defined on a poset [26]. Generalized Lambek Calculus GL involves a finite number of basic operations (with an arbitrary number of arguments) and their residual operations [7]. In this paper we study a further generalization of GL which admits operations whose arguments and values can be of different sorts. This logic is called Multi-Sorted Lambek Calculus mL. We also consider its variants with lattice and boolean operations. We discuss some basic properties of these logics (completeness, decidability, complexity and others) and the corresponding algebras.