Non-associative Algebra - Algebras Satisfying Identities

Algebras Satisfying Identities

Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study. For this reason, the best-known kinds of non-associative algebras satisfy identities which simplify multiplication somewhat. These include the following identities.

In the list, x, y and z denote arbitrary elements of an algebra.

• Associative: (xy)z = x(yz).
• Commutative: xy = yx.
• Anticommutative: xy = −yx.
• Jacobi identity: (xy)z + (yz)x + (zx)y = 0.
• Jordan identity: (xy)x2 = x(yx2).
• Power associative: For all x, and any three nonnegative powers of x associate. That is if a, b and c are nonnegative powers of x, then a(bc) = (ab)c. This is equivalent to saying that xm xn = xn+m for all non-negative integers m and n.
• Alternative: (xx)y = x(xy) and (yx)x = y(xx).
• Flexible: x(yx) = (xy)x.

These properties are related by

1. associative implies alternative implies power associative;
2. associative implies Jordan identity implies power associative;
3. Each of the properties associative, commutative, anticommutative, Jordan identity, and Jacobi identity individually imply flexible.
4. For a field with characteristic not two, being both commutative and anticommutative implies the algebra is just {0}.