Glossary of Order Theory - U

U

• Unit. The greatest element of a poset P can be called unit or just 1 (if it exists). Another common term for this element is top. It is the infimum of the empty set and the supremum of P. The dual notion is called zero.

• Up-set. See upper set.

• Upper bound. An upper bound of a subset X of a poset P is an element b of P, such that x ≤ b, for all x in X. The dual notion is called lower bound.

• Upper set. A subset X of a poset P is called an upper set if, for all elements x in X and p in P, x ≤ p implies that p is contained in X. The dual notion is called lower set.