Maximal element

The terms maximum element and minimum element are precisely used in set theory in the theory of order.

An element of an ordered set is maximal if there is no greater. It is minimal when there is no smaller.

In a totally ordered set, the terms maximal element and greatest element, and minimal element and the smallest element match. A maximal or minimal element of a partially ordered set is not automatically the largest or smallest element.

Definitions

Is a quasi-ordering, a subset of the ground set and.

Examples

  • M = { 2, 3, 4, 6, 9, 12, 18} is the set of non-trivial divisor of the natural number 36 this quantity with respect to the divisibility partially ordered. Minimum elements 2 and 3, a maximum of 12 and 18 There is no smallest and no largest element. Among the integer dividers of nontrivial 36 are 2,3, 2 ​​and 3 minimal, while 12.18, -12 and -18 are maximal.
  • The non-empty subsets of a given non-empty set X are partially ordered by inclusion. Minimum in this order are all one-element subsets { x}, maximum (and biggest ) is X itself
  • In a vector space is a basis of a (with respect to inclusion) maximal linearly independent subset.

Properties

  • Every finite non-empty ordered set has minimal and maximal elements, infinite ordered sets need not have maximal and minimal elements.
  • A totally ordered set has at most one maximum and a minimum element partially ordered sets may have several maximum and minimum elements.
  • If x is the largest element of M, then x is also the only maximal element of M. The converse is not true: even if M has exactly one maximal element, this is not automatically greatest element.
  • If x is the smallest element of M, then x is the only minimal element of M. The converse is not true: even if M has exactly one minimal element, this is not automatically the smallest element.
  • Does each chain in a non -empty partially ordered set is an upper bound, then the amount has at least one maximal element. (This is the lemma of anger. )
  • For two maximum or minimum two elements and is. For partially ordered sets, this means that different maximum and minimum elements are not comparable. This can be further generalized: The set of all maximal elements is an antichain in the order. The same applies to the set of all minimal elements.
Zorn's lemma
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