Title: Proof-search in implicative linear logic as a matching problem. Authors: Philippe de Groote Abstract: We reduce the provability of fragments of multiplicative linear logic to matching problems consisting in finding a one-one-correspondence between two sets of first-order terms together with a unifier that equates the corresponding terms. According to the kind of structure to which these first-order terms belong (free commutative monoid, free monoid, or free groupoid) our matching problem corresponds to provability in the implicative fragment of multiplicative linear logic, in the Lambek calculus, or in the non-associative Lambek calculus. It is hoped that this reduction will give a new insight into the complexity of the Lambek calculus.