Let (X,≤)(X,\leq)(X,≤) be a poset and let S⊆XS\subseteq XS⊆X. The supremum of this subset, if it exists, is an upper bound bbb of SSS in XXXsuch that for all upper bounds xxx of SSS in XXX, b≤xb\leq xb≤x. That is, the supremum is the least upper bound.