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

  We fix a dualizing object C and write A* = C^A. The intent:
maps <X, Y, ..., Z> → A* = C^A should amount to maps <X, Y, ..., Z, A> → C.
  In particular, there is ev_A: <A*, A> → C corresponding to id_A*: A* → A* .
  From A_i , μ: <A₁, ..., A_n> → A₀, Arens’s μ^‡: <(A₁)**, ..., (A_n)**> → (A₀)**
is the last of n+1 2-step †-stages: the steps of †-stage 1 are: (1) the compositum
ev_A0·<id_A0*, μ>: <(A₀)*, A₁, ..., A_n–1, A_n> → <(A₀)*, A₀> → C (step 1) and
(2) the associated multinear μ^†: <(A₀)*, A₁, ..., A_n–1> → (A_n)* (step 2).