Aczélian n-ary semigroups

We show that real continuous, symmetric, and cancellative n-ary semigroups are topologically order-isomorphic to additive real n-ary semigroups. The binary case (n=2) was originally proved by Aczél (Bull. Soc. Math. Fr. 76:59–64, 1949); there symmetry was redundant.