The associativity equation revisited

R. Craigen, Z. Páles

Research output: Contribution to journalArticle

45 Citations (Scopus)


Consideration of the Associativity Equation, x {ring operator} (y {ring operator} z) = (x {ring operator} y) {ring operator} z, in the case where {ring operator}: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, + ∞) - where b = 0 or -1 and a = 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.

Original languageEnglish
Pages (from-to)306-312
Number of pages7
JournalAequationes Mathematicae
Issue number2-3
Publication statusPublished - Jun 1 1989



  • AMS (1980) subject classification: Primary 39B40, Secondary 22A05, 06F05

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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