lub

Lub is an abbreviation for the least upper bound, a fundamental concept in order theory. Given a subset S of a partially ordered set P, an element u of P is an upper bound of S if s ≤ u for every s in S. The least upper bound (lub) is an upper bound that is smallest among all upper bounds; equivalently, u is an upper bound and for any upper bound v, u ≤ v. If the lub belongs to S, then it is also the maximum of S; however, the lub need not lie in S.

In the real numbers with the usual order, every nonempty subset that is bounded above has a lub, known as the supremum. For example, for S = {1, 2, 2.5}, lub(S) = 2.5. For S = (0,1), lub(S) = 1, even though 1 is not in S. If S is unbounded above, it has no lub in the real numbers (though in the extended real line its lub is +∞). The dual concept is the greatest lower bound, or infimum (glb).

Notation and contexts: lub is common in analysis and lattice theory; many textbooks use sup(S) for the same concept. In ordered sets that are not complete, a lub may fail to exist.

Applications include defining convergence, bounds for sequences, optimization, integration, and lattice-based reasoning in computer science. The lub concept helps formalize how a set can be contained under a minimal “ceiling” compatible with its order relations.