poset

• A set \(P\) with a partial order \(\leq\). The order is partial because some elements are not comparable.
• A chain is a subset of \(P\) in which all the elements are comparable.
• A maximal element is an element that nothing else is greater than. There can be many maximal elements. An element that is not comparable with anything else will be a maximal element.
• Given a subset of \(S\), an upper bound of \(S\) is an element of \(P\) that is greater than all other elements in \(S\). An upper bound must be comparable with all other elements in \(S\). The upper bound need not be in \(S\)