Definition:Total Preordering

Definition

Let $S$ be a set.

Let $\precsim$ be a preordering on $S$.

Then $\precsim$ is a total preordering on $S$ iff $\precsim$ is connected.

That is, iff there is no pair of elements of $S$ which is non-comparable:

$\forall x, y \in S: x \precsim y \lor y \precsim x$

Some sources call this a linear preordering.