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#::<math>P_i^k(x_1,\ldots,x_k) \ \stackrel{\mathrm{def}}{=}\ x_i \, .</math>
Operators (the [[___domain of a function]] defined by an operator is the set of the values of the arguments such that every [[function application]] that must be done during the computation provides a well-defined result):
# ''Composition operator'' <math>\circ\,</math> (also called the ''substitution operator''): Given an m-ary function <math>h(x_1,\ldots,x_m)\,</math> and m k-ary functions <math>g_1(x_1,\ldots,x_k),\ldots,g_m(x_1,\ldots, x_k)</math>:
#::<math>h \circ (g_1, \ldots, g_m) \ \stackrel{\mathrm{def}}{=}\ f, \quad\text{where}\quad f(x_1,\ldots,x_k) = h(g_1(x_1,\ldots,x_k),\ldots,g_m(x_1,\ldots,x_k)).</math>
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The ''[[strong equality]]'' relation <math>\simeq</math> can be used to compare partial μ-recursive functions. This is defined for all partial functions ''f'' and ''g'' so that
:<math>f(x_1,\ldots,x_k) \simeq g(x_1,\ldots,x_l)</math>
holds [[if and only if]] for any choice of arguments either both functions are defined and their values are equal or both functions are undefined.
==Examples==
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:: S(a) = a +1 =<sub>def</sub> a', where 1 =<sub>def</sub> 0', 2 =<sub>def</sub> 0 ' ', etc.
* ''Identity function'': Kleene (1952) uses " U{{su|b=i|p=n}} " to indicate the [[identity function]] over the variables x<sub>i</sub>; B-B-J use the identity function id{{su|b=i|p=n}} over the variables x<sub>1</sub> to x<sub>n</sub>:
: U{{su|b=i|p=n}}( '''x''' ) = id{{su|b=i|p=n}}( '''x''' ) = x<sub>i</sub>
: e.g. U{{su|b=3|p=7}} = id{{su|b=3|p=7}} ( r, s, t, u, v, w, x ) = t
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