### Abstract

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 language | English |
---|---|

Pages (from-to) | 306-312 |

Number of pages | 7 |

Journal | Aequationes Mathematicae |

Volume | 37 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Jun 1989 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Aequationes Mathematicae*,

*37*(2-3), 306-312. https://doi.org/10.1007/BF01836453

**The associativity equation revisited.** / Craigen, R.; Páles, Z.

Research output: Contribution to journal › Article

*Aequationes Mathematicae*, vol. 37, no. 2-3, pp. 306-312. https://doi.org/10.1007/BF01836453

}

TY - JOUR

T1 - The associativity equation revisited

AU - Craigen, R.

AU - Páles, Z.

PY - 1989/6

Y1 - 1989/6

N2 - 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.

AB - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=0010762680&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010762680&partnerID=8YFLogxK

U2 - 10.1007/BF01836453

DO - 10.1007/BF01836453

M3 - Article

VL - 37

SP - 306

EP - 312

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

SN - 0001-9054

IS - 2-3

ER -