Definition:Preordering/Partial vs. Total

Preordering: Partial vs. Total
Note that this definition of preordering does not demand that every pair of elements of $S$ is related by $\precsim$.

The way we have defined a preordering, they may be, or they may not be, depending on the context.

If it is the case that $\precsim$ is a connected relation, that is, that every pair of elements is related by $\precsim$, then $\precsim$ is called a total preordering.

If it is specifically not the case that $\precsim$ is connected, then $\precsim$ is called a partial preordering.