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

  In a monoidal setting, there’s a ⊗^‡: <X**, Y**, ..., Z**> → (X ⊗ Y ⊗ ... ⊗ Z)** arising from the universal multilinear ⊗: < X, Y, ..., Z> → X ⊗ Y ⊗ ... ⊗ Z and, when μ: <X, Y, ..., Z> → A, we have (naturality in A) μ^‡ = μ** · ⊗^‡: <X**, Y**, ..., Z**> → (X ⊗ Y ⊗ ... ⊗ Z)** → A**.
  One may also view ⊗^‡ and μ^‡ as maps X** ⊗ Y** ⊗ ... ⊗ Z** → (X ⊗ Y ⊗ ... ⊗ Z)** and X** ⊗ Y** ⊗ ... ⊗ Z** → A** — i.e., Arens’s work exhibits monoidal structure on ** .