  In previous work we showed that the contact category algebra of a
quadrangulated surface is isomorphic to the homology of a strand algebra from
bordered sutured Floer theory. Being isomorphic to the homology of a
differential graded algebra, this contact category algebra has an A-infinity
structure, allowing us to combine contact structures not just by gluing, but
also by higher-order operations.
  In this paper we investigate such A-infinity structures and higher order
operations on contact structures. We give explicit constructions of such
A-infinity structures, and establish some of their properties, including
conditions for the vanishing and nonvanishing of A-infinity operations. Along
the way we develop several related notions, including a detailed consideration
of tensor products of strand diagrams.
