The associativity equation revisited |
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Authors: | R. Craigen Z. Páles |
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Affiliation: | (1) Department of Pure Mathematics, University of Waterloo, N2L 3G1 Waterloo, Ontario, Canada;(2) Inst. of Mathematics, L. Kossuth University, Pf. 12, H-4010 Debrecen, Hungary |
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Abstract: | 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. |
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Keywords: | Primary 39B40 Secondary 22A05, 06F05 |
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