In mathematics, especially in orer theory, an upper bound of a subset S of some partially ordered set (P, ≤) is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound. The empty subset Φ of a partially ordered set P is conventionally considered to be both bounded from above and bounded from below with every element of P being both upper and lower bound of Φ.