Dear All, I wonder if the following notion has already a name and disscussed somewhere: It is like a monoidal category but the associator and units are not invertible. (Lax monoidal categories share this property but they seem to treat the units differently.) It has left and right versions, the "right-monoidal" category consists of a category C, a functor C x C --> C, <M,N> |--> M*N, an object R and natural transformations gamma_L,M,N: L*(M*N) --> (L*M)*N eta_M: M --> R*M eps_M: M*R -->M satisfying 5 axioms (1 pentagon, 3 triangles and eps_R o eta_R = R) that are obtained from the usual monoidal category axioms by expressing everything in terms of the associator, the right unit (eps), and the inverse left unit (eta) never using their inverses. I find this structure interesting because of the following: Thm: Let R be a ring. Closed right-monoidal structures on the category M_R of right R-modules are (up to approp. isomorphisms on both sides) precisely the right R-bialgebroids. (The ordinary monoidal structure remains hidden in the special nature of M_R.) I would thank for any suggestion. Kornel Szlachanyi [For admin and other information see: http://www.mta.ca/~cat-dist/ ]