Editing Maximum and minimum (section)

==In relation to sets==
Maxima and minima can also be defined for sets. In general, if an [[ordered set]] ''S'' has a [[greatest element]] ''m'', then ''m'' is a [[maximal element]] of the set, also denoted as <math>\max(S)</math>. Furthermore, if ''S'' is a subset of an ordered set ''T'' and ''m'' is the greatest element of ''S'' with (respect to order induced by ''T''), then ''m'' is a [[supremum|least upper bound]] of ''S'' in ''T''. Similar results hold for [[least element]], [[minimal element]] and [[infimum|greatest lower bound]]. The maximum and minimum function for sets are used in [[database]]s, and can be computed rapidly, since the maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self-[[decomposable aggregation function]]s.

In the case of a general [[partial order]], a '''least element''' (i.e., one that is less than all others) should not be confused with the '''minimal element''' (nothing is lesser). Likewise, a '''[[greatest element]]''' of a [[partially ordered set]] (poset) is an [[upper bound]] of the set which is contained within the set, whereas the '''maximal element''' ''m'' of a poset ''A'' is an element of ''A'' such that if ''m'' ≤ ''b'' (for any ''b'' in ''A''), then ''m'' = ''b''. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements.  If a poset has more than one maximal element, then these elements will not be mutually comparable.

In a [[total order|totally ordered]] set, or ''chain'', all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element.  Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms '''''minimum''''' and '''''maximum'''''. 

If a chain is finite, then it will always have a maximum and a minimum.  If a chain is infinite, then it need not have a maximum or a minimum.  For example, the set of [[natural number]]s has no maximum, though it has a minimum. If an infinite chain ''S'' is bounded, then the [[topological closure|closure]] ''Cl''(''S'') of the set occasionally has a minimum and a maximum, in which case they are called the '''greatest lower bound''' and the '''least upper bound''' of the set ''S'', respectively.