C.M.S. • Fredericton, NB • June 2010
Meditations on Arens Multiplication
Next    F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus
Commutativity considerations
  For A a monoid in symmetric monoidal category, write • (not μ) for its bilinear multiplication map A × AA (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 × AA (a, b) ··> ab ab = •(a, b)
: A* × AA* (φ, a) ··> [b ··> φ(ab) = (φ • a)(b)] φ • a = φ(a • –)
††: A** × A* → A* (T, φ) ··> [a ··> T(φ • a) = (T†† φ)(a)] T†† φ = T(φ • –)
= •†††: A** × A** → A** (S, T) ··> [φ ··> S(T†† φ) = (S††† T)(φ)] S T = S(T†† –)
  Thus, (eva T)(φ) = eva(T†† φ) = (T†† φ)(a) = T(φ • a) = T(φ(a • –)), and (S evb)(φ) = S(evb†† φ) = S(φ(– • b)).