Editing Function composition (section)

==Multivariate functions==
Partial composition is possible for [[multivariate function]]s. The function resulting when some argument {{math|''x''<sub>''i''</sub>}} of the function {{mvar|f}} is replaced by the function {{mvar|g}} is called a composition of {{mvar|f}} and {{mvar|g}} in some computer engineering contexts, and is denoted {{math|1=''f'' {{!}}<sub>''x''<sub>''i''</sub> = ''g''</sub>}}
<math display="block">f|_{x_i = g} = f (x_1, \ldots, x_{i-1}, g(x_1, x_2, \ldots, x_n), x_{i+1}, \ldots, x_n).</math>

When {{mvar|g}} is a simple constant {{mvar|b}}, composition degenerates into a (partial) valuation, whose result is also known as [[Restriction (mathematics)|restriction]] or ''co-factor''.<ref name="Bryant_1986"/>

<math display="block">f|_{x_i = b} = f (x_1, \ldots, x_{i-1}, b, x_{i+1}, \ldots, x_n).</math>

In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of [[primitive recursive function]]. Given {{mvar|f}}, a {{mvar|n}}-ary function, and {{mvar|n}} {{mvar|m}}-ary functions {{math|''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>}}, the composition of {{mvar|f}} with {{math|''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>}}, is the {{mvar|m}}-ary function
<math display="block">h(x_1,\ldots,x_m) = f(g_1(x_1,\ldots,x_m),\ldots,g_n(x_1,\ldots,x_m)).</math>

This is sometimes called the '''generalized composite''' or '''superposition''' of ''f'' with {{math|''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>}}.<ref name="Bergman_2011"/> The partial composition in only one argument mentioned previously can be instantiated from this more general scheme by setting all argument functions except one to be suitably chosen [[projection function]]s. Here {{math|''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>}} can be seen as a single vector/[[tuple]]-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition.<ref name="Tourlakis_2012"/>

A set of finitary [[operation (mathematics)|operation]]s on some base set ''X'' is called a [[clone (algebra)|clone]] if it contains all projections and is closed under generalized composition. A clone generally contains operations of various [[arity|arities]].<ref name="Bergman_2011"/> The notion of commutation also finds an interesting generalization in the multivariate case; a function ''f'' of arity ''n'' is said to commute with a function ''g'' of arity ''m'' if ''f'' is a [[homomorphism]] preserving ''g'', and vice versa, that is:<ref name="Bergman_2011"/>
<math display="block">f(g(a_{11},\ldots,a_{1m}),\ldots,g(a_{n1},\ldots,a_{nm})) = g(f(a_{11},\ldots,a_{n1}),\ldots,f(a_{1m},\ldots,a_{nm})).</math>

A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity) operation. A binary (or higher arity) operation that commutes with itself is called [[Medial magma|medial or entropic]].<ref name="Bergman_2011"/>