\begin{definition}[Definition:Operator Domain]
An operator domain is a notational tool for indexing a collection of operations on a set.
Formally, an '''operator domain''' is a set $\Omega$  together with a mapping $a : \Omega \to \N_0$.
We also use the notation, for $n \in \N_0$, $\Omega(n) = \{\omega \in \Omega : a(\omega) = n \}$.
The elements of $\Omega$ are '''operators'''.
If $\omega \in \Omega$, then $a(\omega)$ is the '''arity''' of the operator $\omega$.
The elements of $\Omega(n)$ are the '''$n$-ary''' operators of $\Omega$.
Note that an operator domain is an abstract labelling set, so the definitions here are purely formal; we have not identified any concrete operations.
The latter is done using an Omega algebra.
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\end{definition}