Two axiomatizations of the nonassociative and commutative Lambek syntactic calculus are given and their equivalence is proved. The first axiomatization employs Permutation as the only structural rule, the second one, with no Permutation rule, employs only unidirectional types. It is also shown that in the case of the Ajdukiewicz calculus an analogous equivalence is valid only in the case of a restricted set of formulas. Unidirectional axiomatizations are employed in order to establish the generative power of categorial grammars based on (...) the nonassociative and commutative Lambek calculus with product. Those grammars produce CF-languages of finite degree generated by CF-grammars closed with respect to permutations. (shrink)