Family Operations

Why

We want to generalize operations beyond two objects.

Operations

The pairwise extension of a commutative operation is the function from finite families of the ground set to the ground setobtained by applying the operation pairwise to elements. TODO: this is not a function if the operation is not commutative.

The ordered pairwise extension of an operation is the function from finite families ground set to the ground set obtained by applying the operation pairwise to elements in order.

Notation

Let $(A, +)$ be an algebra and $\set{A_{i}}_{i=1}^{n}$ a finite family of elements of $A$. We denote the pairwise extension by

\[ \overset{n}{\underset{i=1}{+}} A_i \]