Definition:Total Preordering

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.