Multigrade operator

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A multigrade operator ${\displaystyle \Omega \!}${\displaystyle \Omega \!} is a parametric operator with parameter ${\displaystyle k\!}${\displaystyle k\!} in the set ${\displaystyle \mathbb {N} \!}${\displaystyle \mathbb {N} \!} of non-negative integers.

The application of a multigrade operator ${\displaystyle \Omega \!}${\displaystyle \Omega \!} to a finite sequence of operands ${\displaystyle (x_{1},\ldots ,x_{k})\!}${\displaystyle (x_{1},\ldots ,x_{k})\!} is typically denoted with the parameter ${\displaystyle k\!}${\displaystyle k\!} left tacit, as the appropriate application is implicit in the number of operands listed. Thus ${\displaystyle \Omega (x_{1},\ldots ,x_{k})\!}${\displaystyle \Omega (x_{1},\ldots ,x_{k})\!} may be taken for ${\displaystyle \Omega _{k}(x_{1},\ldots ,x_{k}).\!}${\displaystyle \Omega _{k}(x_{1},\ldots ,x_{k}).\!}

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