Let denote an -algebra, so that is a Vector Space over and
where is vector multiplication which is assumed to be Bilinear. Now define
where . is said to be tame if is a finite union of Subspaces of . A 2-D 0-Associative
algebra is tame, but a 4-D 4-Associative algebra and a 3-D 1-Associative algebra need not be tame. It is
conjectured that a 3-D 2-Associative algebra is tame, and proven that a 3-D 3-Associative algebra is tame
if it possesses a multiplicative Identity Element.
Finch, S. ``Zero Structures in Real Algebras.''
© 1996-9 Eric W. Weisstein