Supremum

A sublattice of a lattice is a subset that is closed under pairwise infima and suprema.

The best way to describe the supremum of S is to say that it wants to be the greatest element of S. In fact, if S has a greatest element, then that element is the supremum.

The key to an approach to vector optimization based on infimum and supremum is to consider set-based objective functions and to extend the partial ordering of the original objective space to a suitable subspace of the power set. In this new space the infimum and supremum exist under the usual assumptions.