Definition:Ordering/Partial vs. Total
Ordering: Partial vs. Total
It is not demanded of an ordering $\preceq$, defined in its most general form on a set $S$, that every pair of elements of $S$ is related by $\preceq$.
They may be, or they may not be, depending on the specific nature of both $S$ and $\preceq$.
If it is the case that $\preceq$ is a connected relation, that is, that every pair of distinct elements is related by $\preceq$, then $\preceq$ is called a total ordering.
If it is not the case that $\preceq$ is connected, then $\preceq$ is called a partial ordering.
Beware that some sources use the word partial for an ordering which may or may not be connected, while others insist on reserving the word partial for one which is specifically not connected.
It is wise to be certain of what is meant.
As a consequence, on $\mathsf{Pr} \infty \mathsf{fWiki}$ we resolve any ambiguity by reserving the terms for the objects in question as follows:
- Partial ordering: an ordering which is specifically not total
- Total ordering: an ordering which is specifically not partial.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): partial order
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): partial order