# Some Arithmetical Semigroups

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

## Abstract

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.

Argentina

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© 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