Editing Associative property (section)

== Definition ==
[[File:Semigroup_associative.svg|thumbnail|A binary operation ∗ on the set ''S'' is associative when [[Commutative diagram|this diagram commutes]]. That is, when the two paths from {{math|{{var|S}}×{{var|S}}×{{var|S}}}} to {{mvar|S}} [[Function composition|compose]] to the same function from {{math|{{var|S}}×{{var|S}}×{{var|S}}}} to {{mvar|S}}.]]
Formally, a [[binary operation]] <math>\ast</math> on a [[Set (mathematics)|set]] {{mvar|S}} is called '''associative''' if it satisfies the '''associative law''':
:<math>(x \ast y) \ast z = x \ast (y \ast z)</math>, for all <math>x,y,z</math> in {{mvar|S}}.

Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol ([[Juxtaposition#Mathematics|juxtaposition]]) as for [[multiplication]].
:<math>(xy)z = x(yz)</math>, for all <math>x,y,z</math> in {{mvar|S}}.

The associative law can also be expressed in functional notation thus: <math>(f \circ (g \circ h))(x) = ((f \circ g) \circ h)(x)</math>