Let \((X,\leq)\) be a partially ordered set. A chain in \(X\) is a subset \(C\subseteq X\) that is totally ordered by \(\leq\): for all \(c,c'\in C\), \(c\leq c'\) or \(c'\leq c\).
Let \((X,\leq)\) be a partially ordered set and \(S\subseteq X\). An upper bound of \(S\) is an element \(x\in X\) such that \(s\leq x\) for all \(s\in S\).