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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.2012-05-02
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    Based on a glance at the article, I say "no, they are not related."2012-05-02
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    @Yrogirg: Perhaps you meant to ask about [composition rings](http://en.wikipedia.org/wiki/Composition_ring).2012-05-02

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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.