Let denote an -algebra, so that is a vector space over and
(1)
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(2)
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where is vector multiplication which is assumed to be bilinear. Now define
(3)
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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.