Definition:Operator Domain

Definition
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.