Definition:Total Preordering

Definition
Let $S$ be a set.

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

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

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


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

Also known as
Some sources call this a linear preordering.

Some sources within the field of game theory call it a preference relation, in accordance with its specific application.