Let 
denote an 
-algebra, so that 
is a vector space over 
and
 |
(1)
|
 |
(2)
|
where 
is vector multiplication which is assumed
to be bilinear. Now define
 |
(3)
|
where 
.
is said to be tame if 
is a finite union of subspaces
of 
. A two-dimensional 0-associative
algebra is tame, but a four-dimensional 4-associative
algebra and a three-dimensional 1-associative algebra
need not be tame. It is conjectured that a three-dimensional 2-associative
algebra is tame, and proven that a three-dimensional 3-associative
algebra is tame if it possesses a multiplicative identity
element.
