Total and Partial Ordering

Determining the order of events is a common problem that needs to be solved in software systems.

However, there are two different possible types of ordering: total ordering and partial ordering.

Total ordering

A total order is a binary relation that compares any two elements of a set with each other. As a direct consequence of this property, we can use this relation to derive only a single order for all the elements in a set. This is why it’s called a total order.

Example

If we totally order the set of unique integers $\lbrace 7, 9, 3, 2, 6 \rbrace$ using the relation less than $<$, the associated total order is $[2, 3, 6, 7, 9]$.