Preliminaries to the Lambek Calculus

The articles [16], [9], [19], [18], [13], [1], [6], [21], [5], [10], [12], [17], [20], [7], [8], [15], [11], [2], [3], and [4] provide the notation and terminology for this paper. We consider structures of the type algebra as extensions of 1-sorted structure as systems 〈 a carrier, a left quotient, a right quotient, an inner product 〉, where the carrier is a set and the left quotient, the right quotient, and the inner product are binary operations on the carrier. One can check that there exists a structure of the type algebra which is non empty and strict. Let s be a non empty structure of the type algebra. A type of s is an element of s. We follow the rules: s denotes a non empty structure of the type algebra, T , X , Y denote finite sequences of elements of the carrier of s, and x, y, z denote types of s. Let us consider s, x, y. The functor x\ y yields a type of s and is defined by: