Tensor term logic for categorial grammars: simple unification of commutative and noncommutative structure

A prototypical example of categorial grammars are those based on Lambek calculus, i.e. noncommutative intuitionistic linear logic. However, it has been noted that purely noncommutative operations are often not sufficient for modeling even very simple natural language phenomena. Therefore a number of alternative formalisms are considered in the literature: those using purely ``commutative'' linear logic as well as combining (to some level) commutative and non-commutative operations. The logic of tensor terms that we propose is a variant of such a combination. This logical calculus was designed specially for defining categorial grammars. It contains Lambek calculus and multiplicative linear logic as conservative fragments, yet the syntax is very simple, not departing much from that of multiplicative linear logic. The system is cut-free and decidable, it has both intuitionistic and classical versions, besides it is equipped with a simple intuitive semantics, which is sound and complete.