Category:Definitions/Partial Orderings

This category contains definitions related to Partial Orderings.
Related results can be found in Category:Partial Orderings.

Let $\struct {S, \preceq}$ be an ordered set.

Then the ordering $\preceq$ is a partial ordering on $S$ if and only if $\preceq$ is not connected.

That is, if and only if $\struct {S, \preceq}$ has at least one pair which is non-comparable:

$\exists x, y \in S: x \npreceq y \land y \npreceq x$