supremum

1. (set theory) (real analysis): Given a subset X of R, the smallest real number that is ≥ every element of X; (order theory): given a subset X of a partially ordered set P (with partial order ≤), the least element y of P such that every element of X is ≤ y.Examples: "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."Synonyms: least upper bound, LUB, sup