Acz\'elian n-ary semigroups

TL;DR

This paper proves that real continuous, symmetric, and cancellative n-ary semigroups are topologically order-isomorphic to additive real n-ary semigroups, extending Aczél's binary case to higher arity.

Contribution

It generalizes Aczél's binary semigroup result to n-ary semigroups, establishing a topological isomorphism under symmetry and cancellativity.

Findings

Real continuous, symmetric, cancellative n-ary semigroups are topologically order-isomorphic to additive n-ary semigroups.

Extension of Aczél's binary case to n-ary semigroups.

Symmetry was redundant in the binary case but essential in the n-ary generalization.

Abstract

We show that the 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 Acz\'el (1949); there symmetry was redundant.