In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. A bounded poset is a poset that has both a greatest element and a least element. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ...

Formally, given a partially ordered set (P, ≤), then an element g of a subset S of P is the greatest element of S if

s ≤ g, for all elements s of S.

Hence, the greatest element of S is an upper bound of S that is contained within this subset. It is necessarily unique. By using ≥ instead of ≤ in the above definition, one defines the least element of S. In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ...

Like upper bounds, greatest elements may fail to exist. Even if a set has some upper bounds, it need not have a greatest element, as the example of the real numbers strictly smaller than 1 shows. This also demonstrates that the existence of a least upper bound (the number 1 in this case) does not imply the existence of a greatest element either. Similar conclusions hold for least elements. The text or formatting below is generated by a template which has been proposed for deletion. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ...

Greatest elements of a partially ordered subset must not be confused with maximal elements of such a set. The difference is discussed in the article on maximal elements. However, in some special cases, such as when dealing with totally ordered sets, both terms do indeed coincide. In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

The least and greatest elements of the whole partially ordered set play a special role and are also called bottom and top or zero (0) and unit (1), respectively. The later notation of 0 and 1 is only used when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1. The existence of least and greatest elements is a special completeness property of a partial order. In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ...

Further introductory information is found in the article on order theory. Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...