C.M.S. • Fredericton, NB • June 2010
Meditations on Arens Multiplication
   F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus

  For A a monoid in symmetric monoidal category, write • (not μ) for its bilinear multiplication map A × A → A (Arens’s suggestive/ambiguous product notation).
  Using a and b as variables from A, φ as variable from A*, and S and T as variables from A**, we have the following stages towards the Arens multiplication on A**:

•: A × A → A	(a, b) ··> a • b	a • b = •(a, b)
•†: A* × A → A*	(φ, a) ··> [b ··> φ(a • b) = (φ •† a)(b)]	φ •† a = φ(a • –)