Grammars based on variants of the Lambek calculus

We consider the Lambek calculus with one division and one primitive type (L(p)) and two extensions of the Lambek calculus: L1 (adding the unit constant) and LR (adding a unary connective that corresponds to taking the language of the inverse strings). We prove that L(p)- grammars and LR-grammars generate precisely all context-free languages without the empty word, and L1-grammars generate precisely all context-free languages.