First order linear logic and tensor type calculus for categorial grammars

We study relationship between first order multiplicative linear logic (MLL1), which has been known to provide representations to different categorial grammars, and the recently introduced extended tensor type calculus (ETTC). We identify a fragment of MLL1, which seems sufficient for many grammar representations, and establish a correspondence between ETTC and this fragment. The system ETTC, thus, can be seen as an alternative syntax and intrinsic deductive system together with a geometric representation for the latter. We also give a natural deduction formulation of ETTC, which might be convenient.