Definition:Partial Preordering
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Definition
Let $S$ be a set.
Let $\precsim$ be a preordering on $S$.
Then $\precsim$ is a partial preordering on $S$ if and only if $\precsim$ is not connected.
That is, if and only if there is at least one pair of elements of $S$ which is non-comparable:
- $\exists x, y \in S: x \not \precsim y \land y \not \precsim x$