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Mathematics Start‑class Low‑priority

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Noncommutative versus Not-necessarily commutative

There is a perpetual problem with terms like non-commutative and non-associative, in that we usually intend to include objects that actually are commutative, associative, or whatever. There's another problem here in that Jordan algebras are commutative by definition, so that there is no such thing as a noncommutative Jordan algebra taken literally.

I would not expand the introduction beyond all reasonable limits so suggest some wording such as

A noncommutative Jordan algebra is a generalisation of a Jordan algebra in which the condition of commutativity is replaced by a weaker condition. The definition includes Jordan algebras as a special case.

That still seems clumsy. Any better suggestions? Hyperbaric oxygen (talk) 15:20, 7 May 2013 (UTC)

Defining axioms

The definition in the 1955 paper of Schafer is not the same as that in the article. Schafer begins by defining a noncommutative Jordan algebra as satisfying the Jordan condition (1)

${\displaystyle (x^{2}a)x=x^{2}(ax)}$

together with the flexible condition (4)

${\displaystyle (xa)x=x(ax)}$

Indeed, Schafer notes that this is also the definition of Albert in 1948 (which Albert found unsatisfying!). Some discussion needs to be made about the equivalence of the definitions in the article and in the references: Schafer notes that under the assumption of (4) the condition (1) is equivalent to any of the conditions (5) of the article. Hyperbaric oxygen (talk) 15:20, 7 May 2013 (UTC)

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