The Shuue Hopf Algebra and Noncommutative Full Completeness

We present a full completeness theorem for the multiplicative fragment of a variant of non-commutative linear logic known as cyclic linear logic (CyLL), rst deened by Yetter. The semantics is obtained by considering dinatural transformations on a category of topological vector spaces which are equivariant under certain actions of a noncocommutative Hopf algebra, called the shuue algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that the space has the denotations of cut-free proofs in CyLL+MIX as a basis. This can be viewed as a fully faithful representation of a free-autonomous category, canonically enriched over vector spaces. This work is a natural extension of the authors' previous work, \Linear LL auchli Semantics", where a similar theorem is obtained for the commutative logic. In that paper, we consider dinaturals which are invariant under certain actions of the additive group of integers. We also present here a simpliication of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras corresponds to the passage from commutative to noncommutative logic.