Definition:Total Preordering

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Let $S$ be a set.

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

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

That is, if and only if 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.