Retrieving "Associativity" from the archives

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1. Function Composition

   Linked via "associative"

   Associativity
   Composition is always associative. For any three composable functions $f: A \to B$, $g: B \to C$, and $h: C \to D$, the following always holds:
   $$
   h \circ (g \circ f) = (h \circ g) \circ f

2. Jordan Algebras

   Linked via "associativity"

   Relationship to Commutators
   The relationship between the Jordan product ($\circ$) and the standard Lie bracket ($[x, y] = xy - yx$) is subtle. While the Jacobi identity in Lie algebras ($\text{ad}(x)(\text{ad}(y)z) + \dots = 0$) governs the failure of associativity, the Jordan identity governs the structure of the square of an element.
   It has been proven that any Jordan algebra $J$ possesses an associated [Lie algebra](/entries/…