= Partially Ordered Set =

The following was adapted from Wolfram's site:

A partially ordered set (or poset) is a set taken together with a partial order  on it. Formally, a partially ordered set is defined as an ordered pair $$P=\left<X,\sqsubseteq \right>$$, where $$X$$ is called the ground set of $$P$$ and $$\sqsubseteq$$ is the partial order of $$P$$.

An element $$u$$ in a partially ordered set $$\left< X,\sqsubseteq \right>$$ is said to be an upper bound for a subset $$S$$ of $$X$$ if for every $$s \in S$$, we have $$s \sqsubseteq u$$. Similarly, a lower bound for a subset $$S$$ is an element $$l$$ such that for every $$s \in S$$, $$l \sqsubseteq s$$. If there is an upper bound and a lower bound for X, then the poset $$\left< X,\sqsubseteq \right>$$ is said to be bounded.


See PartialOrder