  In (Van Benthem, 1991) it is proved that all permutation closures of
context-free languages can be generated by grammars over the Lambek calculus
with the permutation rule (LP-grammars); however, to our best knowledge, it is
not established whether the converse holds or not. In this paper, we show that
LP-grammars are equivalent to linearly-restricted branching vector addition
systems with states and with additional memory (shortly, lBVASSAM), which are
modified branching vector addition systems with states. Then an example of such
an lBVASSAM is presented, which generates a non-semilinear set of vectors; this
yields that LP-grammars generate more than permutation closures of context-free
languages. Moreover, equivalence of LP-grammars and lBVASSAM allows us to
present a normal form for LP-grammars and, as a consequence, prove that
LP-grammars are equivalent to LP-grammars without product. Finally, we prove
that the class of languages generated by LP-grammars is closed under
intersection.
