Жартылай реттелген жиынның ең үлкен және ең кіші элементтері

Element ≥ (or ≤) each other element

In mathematics, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of The term least element is defined dually, that is, it is an element of that is smaller than every other element of

A finite chain always has a greatest and a least element.

The least and greatest element of the whole partially ordered set play a special role and are also called bottom (⊥) and top (⊤), or zero (0) and unit (1), respectively. If both exist, the poset is called a bounded poset. The notation of 0 and 1 is used preferably when the poset is a complemented lattice, and when no confusion is likely, i. e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top. The existence of least and greatest elements is a special completeness property of a partial order. Further introductory information is found in the article on order theory.

The subset of integers has no upper bound in the set of real numbers. Let the relation on be given by The set has upper bounds and but no least upper bound, and no greatest element (cf. picture). In the rational numbers, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound. In the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element. In the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound. In with the product order, the set of pairs with has no upper bound. In with the lexicographical order, this set has upper bounds, e. g. It has no least upper bound.