 In the presentation of the syntactic calculus, the structural options take the form of non-logical axioms (postulates), which one adds to the residuation laws.  In the sequent presentation, one has corresponding structural rules.  The sequent calculus for L(P) also allows for a sugared presentation where the antecedent is treated as a list or a multiset of formulas; the inference steps for the structural rules can then be left implicit. With respect to the frame semantics, each of the structural options introduces a constraint on the interpretation of the Merge relation R⊗.  Completeness for the calculi with structural options is then with respect to all frames respecting the relevant constraints (Došen 1992). The associative calculus L in fact allows more specific models where we read Rxyz as x = y · z, with · a binary concatenation operation.  Pentus (1995) presents a quite intricate proof that L is indeed complete with respect to such free semigroup models (or language models, as they are also called).