It has been proved by the author that the product-free Lambek calculus with the empty string in its associative (L ₀) and non-associative (NL ₀) variant is not finitely Gentzen-style axiomatizable if the only rule of inference is the cut rule. We give here rather detailed outlines of the proofs for both L ₀ and NL ₀. In the last section, Hilbert-style axiomatics for the corresponding weak implicational calculi are given.