Acz\'elian n-ary semigroups
Miguel Couceiro, Jean-Luc Marichal

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.
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