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I've read the definition of composition algebra in wikipedia, but I couldn't understand whether it relates to the usual function composition $(\circ)$.

Are these two things related at all?

Considering functions of type $\mathbb R \to \mathbb R$ will suffice.

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No, the two concepts are not related at all. You may have been mislead by language. In the concept composition algebra, the word 'composition' is an attribute narrowing down the class of algebras (over a field). IOW, it deals about special kind of algebras. English is not my first language, so I don't for sure, but I think that the phrase algebra of compositions would serve as an umbrella concept for the algebraic rules one may encounter when studying compositions of functions. –  Jyrki Lahtonen May 2 '12 at 7:23
    
Based on a glance at the article, I say "no, they are not related." –  anon May 2 '12 at 7:23
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@Yrogirg: Perhaps you meant to ask about composition rings. –  Zhen Lin May 2 '12 at 7:56

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From the comments: no, they are not related. The concept of function composition $\circ$ is embedded into composition ring, not the composition algebra. So yes, it may be quite misleading.

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