Natural Associativity without the Pentagon Condition

A premonoidal category is equipped only with a bifunctor and a natural isomorphism for associativity. We introduce a (deformation) natural automorphism, q, representing the deviation from the Pentagon condition. We uncover a binary tree representation for all diagrams involving a and q and provide a link to permutations and linear orderings. This leads to other notions of premonoidalness. We define these notions and prove coherence results for each. PACS number: 02.10.Ws, 02.20.Qs, 02.20.Fh, 02.20.Df, 03.65.Fd, 31.15.Hz Natural Associativity without the Pentagon Condition 2