Maximum and minimum element

The terms maximum element and minimum element are used in set theory , more precisely in order theory.

An element of an ordered set is maximal if there is no larger one. It's minimal when there isn't a smaller one.

In a totally ordered set, the terms maximum element and largest element as well as minimum element and smallest element match. A maximum or minimum element of a partially ordered set is not automatically its largest or smallest element.

Definitions

${\displaystyle (X,\leq )}$be a quasi-order , a subset of the basic set and . ${\displaystyle M\subseteq X}$ ${\displaystyle X}$ ${\displaystyle x\in M}$

${\displaystyle x\ }$is the maximum element of ${\displaystyle M\ }$ ${\displaystyle :\Longleftrightarrow \forall y\in M:(x\leq y\Rightarrow y\leq x)}$

${\displaystyle x\ }$is a minimal element of ${\displaystyle M\ }$ ${\displaystyle :\Longleftrightarrow \forall y\in M:(y\leq x\Rightarrow x\leq y)}$

Examples

• M  : = {2, 3, 4, 6, 9, 12, 18} is the set of nontrivial natural divisors of the number 36. This set is partially ordered with regard to divisibility . Minimum elements are 2 and 3, maximum are 12 and 18. There is no smallest or largest element. Among the integer nontrivial divisors of 36, 2, 3, −2 and −3 are 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 . All one-element subsets { x } are minimal in this order , the maximal (and also the largest) element is X itself.
• In a vector space a basis is a (with respect to inclusion) maximal linearly independent subset.
• In each ring is due , and thus , a greatest element in terms of divisibility and thus maximum. All units in are the smallest elements and therefore also minimal. ${\displaystyle (R,\mid )}$ ${\displaystyle 0}$ ${\displaystyle \forall r\in R:r\cdot 0=0}$ ${\displaystyle \forall r\in R:r\mid 0}$ ${\displaystyle \mid }$ ${\displaystyle (R,\mid )}$

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 maximal and one minimal element, partially ordered sets can have several maximal and minimal elements.
• Is x the largest element of M , then x also the only maximal element M . If M is finite, then the reverse also applies: If M has exactly one maximum element, then this is also the largest element. This statement does not apply to infinite sets.
• If x is the smallest element of M , then x also the only minimal element of M . If M is finite, then the converse also applies: If M has exactly one minimal element, then this is also the smallest element. This statement does not apply to infinite sets.
• If every chain in a non - empty semi-ordered set has an upper bound , then the set has at least one maximum element. (This is the lemma of anger .)
• For two maximal or two minimal elements and applies . In the case of half orders, this means that different maximum or minimum elements cannot be compared. This can still be generalized: The set of all maximal elements is an anti-chain in the order. The same applies to the set of all minimal elements. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x\leq y\Rightarrow y\leq x}$

literature

• Oliver Deiser: Introduction to set theory, 2nd edition, Springer, Berlin 2004, ISBN 3-540-20401-6