Some Arithmetical Semigroups

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.