The associativity equation revisited

Summary

Consideration of the Associativity Equation,x ∘ (y ∘ z) = (x ∘ y) ∘ z, in the case where∘:I × I → I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( − ∞,b), ( − ∞,b], −∞, +∞), (a, + ∞), or [a, + ∞) — whereb = 0 or −1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Páles and fine-tuned by Craigen.