Talk:Power associativity

The old page history is so corrupted that I can't tell who added the false statement that power-associativity of a magma is equivalent to x(xx) = (xx)x for all x. But anyway, here's a minimal counterexample:

    | a b
  --+-----
  a | b a
  b | a a

This is commutative, so the identity x(xx) = (xx)x obviously holds. But either element generates the whole magma, and this is not a semigroup, as a(ab) = b while (aa)b = a. --Zundark 10:48 Dec 1, 2002 (UTC)

That's my fault; an analogous idea works for alternativity, but not for power associativity. -- Toby 07:11 Feb 21, 2003 (UTC)

Okay, now I'm confused. I always heard associativity defined as x(xx)=(xx)x. So is power assoc. a /stronger/ statement than associativity or weaker? Lunkwill 20:43, 7 Aug 2004 (UTC)

Associativity is x(yz) = (xy)z, which is stronger than power associativity, which is stronger than x(xx) = (xx)x. --Zundark 07:48, 8 Aug 2004 (UTC)

Two very minor quibbles! "Every associative algebra is obviously power-associative, but so too are alternative algebras like the octonions and even some non-alternative algebras like the sedenions" is a perfectly correct statement, but it makes associative and alternative algebras sound distinct. I think a better formulation would be "Every associative algebra is obviously power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even some non-alternative algebras like the sedenions." Also, perhaps for contrast an example should be given of an algebra which isn't power-associative? --VivaEmilyDavies 14:12, 24 Nov 2004 (UTC)

Wait, are we sure that "every alternative algebra" is power-associative? See this counterexample https://math.stackexchange.com/questions/3132318/why-an-alternative-magma-need-not-even-be-power-associative/3132319#3132319. --Mathematician-at-heart (talk) 14:50, 1 March 2022 (UTC)

Yes, every alternative algebra is power-associative. The extra structure in an algebra (compared to a magma) makes a big difference. In fact, it's a theorem of Artin that any pair of elements in an alternative algebra generates an associative subalgebra. This is mentioned (with a reference) in the Properties section of the alternative algebra article. --Zundark (talk) 21:11, 1 March 2022 (UTC)