Some Arithmetical Semigroups

  • A. S. Fraenkel
  • H. Porta
  • K. B. Stolarsky
Part of the Progress in Mathematics book series (PM, volume 85)


The peculiar multiplication rule is asssociative but does not always produce an integer for integers
$$ n\,x\,m\, = \,\left\lfloor {\left( {\frac{3}{2}} \right)\,nm} \right\rfloor $$
n and m. If we try to force the result to be an integer by truncation, i. e., by
$$ n\,x\,m\, = \,\left\lfloor {\left( {\frac{3}{2}} \right)\,nm} \right\rfloor $$
we no longer have an associative multiplication. For example (3 x (5 x 7)) = 234. It would seem exceptional to find associativity in operations whose definition involves truncation. Our object is to study several such exceptional operations.


Triple Product Fibonacci Number Lucas Number Associative Operation Canadian Math 
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Copyright information

© Bikhäuser Boston 1990

Authors and Affiliations

  • A. S. Fraenkel
    • 1
  • H. Porta
    • 2
  • K. B. Stolarsky
    • 1
  1. 1.Dept. of Applied MathematicsThe Weizman Institute of ScienceRehovotIsreal
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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