Definition:Partially Ordered Set

A poset (convenient abbreviation for partially ordered set) is a relational structure $$\left({S; \preceq}\right)$$ such that $$\preceq$$ is a partial ordering.

The poset $$\left({S; \preceq}\right)$$ is said to be partially ordered by $$\preceq$$.

In general, a poset can also be a relational structure $$\left({S; \preceq}\right)$$ such that $$\preceq$$ is an ordering which may or may not be partial.