Maximal element

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.

Definition

Let Failed to parse (Missing texvc executable; please see math/README to configure.): P,\leq

be a partially ordered set and Failed to parse (Missing texvc executable; please see math/README to configure.): S\subset P

, then Failed to parse (Missing texvc executable; please see math/README to configure.): m\in S

is a maximal element of S if 

for all Failed to parse (Missing texvc executable; please see math/README to configure.): s\in S

and Failed to parse (Missing texvc executable; please see math/README to configure.): m \leq s 
imply Failed to parse (Missing texvc executable; please see math/README to configure.):  m = s. 

The definition for minimal elements is obtained by using ≥ instead of ≤.

Existence and uniqueness

Maximal elements need not exist.

Example 1: Let Failed to parse (Missing texvc executable; please see math/README to configure.): S=[1,\infty]\subset \Bbb{R}

, for all Failed to parse (Missing texvc executable; please see math/README to configure.): m\in S

we have Failed to parse (Missing texvc executable; please see math/README to configure.): s=m+1\in S
but Failed to parse (Missing texvc executable; please see math/README to configure.): m<s
(that is, Failed to parse (Missing texvc executable; please see math/README to configure.): m\leq s
but not Failed to parse (Missing texvc executable; please see math/README to configure.): m=s

).

Example 2: Let Failed to parse (Missing texvc executable; please see math/README to configure.): S=\{s\in \Bbb{Q}:1\leq s^{2}\leq 2)\}\subset \Bbb{Q}

and recall that Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2}\notin \Bbb{Q}

.

In general Failed to parse (Missing texvc executable; please see math/README to configure.): \leq

is only a partial order on Failed to parse (Missing texvc executable; please see math/README to configure.): S

. If Failed to parse (Missing texvc executable; please see math/README to configure.): m

is a maximal element and Failed to parse (Missing texvc executable; please see math/README to configure.): s\in S

, it remains the possibility that neither Failed to parse (Missing texvc executable; please see math/README to configure.): s\leq m

nor Failed to parse (Missing texvc executable; please see math/README to configure.): m\leq s

. This leaves open the possibility that there are many maximal elements.

Example 3: Let Failed to parse (Missing texvc executable; please see math/README to configure.): A

be a set with at least two elements and let Failed to parse (Missing texvc executable; please see math/README to configure.): S=\{\{a\}:a\in A\}
be a subset of the power set Failed to parse (Missing texvc executable; please see math/README to configure.): P(A)

, partially ordered by Failed to parse (Missing texvc executable; please see math/README to configure.): \subset . Every element Failed to parse (Missing texvc executable; please see math/README to configure.): \{a\}\in S

is a maximal (and minimal) and for any Failed to parse (Missing texvc executable; please see math/README to configure.): a^{\prime },a^{\prime \prime}
neither Failed to parse (Missing texvc executable; please see math/README to configure.): \{a^{\prime }\} \subset \{a^{\prime \prime}\}
nor Failed to parse (Missing texvc executable; please see math/README to configure.): \{a^{\prime\prime }\} \subset \{a^{\prime }\}

.

Maximal elements and the greatest element

It looks like Failed to parse (Missing texvc executable; please see math/README to configure.): m

should be a greatest element or maximum but in fact it is not necessarily the case: the definition of maximal element is somewhat weaker. Suppose we find Failed to parse (Missing texvc executable; please see math/README to configure.): s\in S
with Failed to parse (Missing texvc executable; please see math/README to configure.): \max S\leq s

, then, by the definition of greatest element, Failed to parse (Missing texvc executable; please see math/README to configure.): s\leq \max S

so that Failed to parse (Missing texvc executable; please see math/README to configure.): s=\max S

. In other words, a maximum, if it exists, is the (unique) maximal element.

The reverse is not true: there can be maximal elements despite there being no maximum. Example 3 is an instance of existence of many maximal elements and no maximum. The reason is, again, that in general Failed to parse (Missing texvc executable; please see math/README to configure.): \leq

is only a partial order on Failed to parse (Missing texvc executable; please see math/README to configure.): S

. If Failed to parse (Missing texvc executable; please see math/README to configure.): m

is a maximal element and Failed to parse (Missing texvc executable; please see math/README to configure.): s\in S

, it remains the possibility that neither Failed to parse (Missing texvc executable; please see math/README to configure.): s\leq m

nor Failed to parse (Missing texvc executable; please see math/README to configure.): m\leq s

.

Of course, when the restriction of Failed to parse (Missing texvc executable; please see math/README to configure.): \leq

to Failed to parse (Missing texvc executable; please see math/README to configure.): S
is a total order, the notions of maximal element and greatest element coincide. Let Failed to parse (Missing texvc executable; please see math/README to configure.): m\in S
be a maximal element, for any Failed to parse (Missing texvc executable; please see math/README to configure.): s\in S
either Failed to parse (Missing texvc executable; please see math/README to configure.): s\leq m
or Failed to parse (Missing texvc executable; please see math/README to configure.): m\leq s

. In the second case the definition of maximal element requires Failed to parse (Missing texvc executable; please see math/README to configure.): m=s

so we conclude that Failed to parse (Missing texvc executable; please see math/README to configure.): x\in B
implies Failed to parse (Missing texvc executable; please see math/README to configure.): s\leq m

. In other words, Failed to parse (Missing texvc executable; please see math/README to configure.): m

is a greatest element.

Finally, let us remark that Failed to parse (Missing texvc executable; please see math/README to configure.): S

being totally ordered is sufficient to ensure that a maximal element is a greatest element, but it is not necessary.

Directed sets

In a totally ordered set, the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. This observation does not only apply to totally ordered subsets of any poset, but also to their order theoretic generalization via directed sets. In a directed set, every pair of elements (especially pairs of incomparable elements) has a common upper bound within the set. It is easy to see that any maximal element of such a subset will be unique (unlike in a poset). Furthermore, this unique maximal element will also be the greatest element.

Similar conclusions are true for minimal elements.

Further introductory information is found in the article on order theory.

Preorder relations in economic theory

There is no reason to limit the notion of maximal element to orderings. However, the terminology changes from one type of relation to the other for reasons we shall see.

In consumer theory the consumption space is some set Failed to parse (Missing texvc executable; please see math/README to configure.): X , usually the positive orthant of some vector space so that each Failed to parse (Missing texvc executable; please see math/README to configure.): x\in X

represents a quantity of consumption specified for each existing commodity in the

economy. Preferences of a consumer are usually represented by a total preorder Failed to parse (Missing texvc executable; please see math/README to configure.): \preceq

so that Failed to parse (Missing texvc executable; please see math/README to configure.): x,y\in X
and Failed to parse (Missing texvc executable; please see math/README to configure.): x\preceq y
reads: Failed to parse (Missing texvc executable; please see math/README to configure.): x
is at most as preferred as Failed to parse (Missing texvc executable; please see math/README to configure.): y

. When Failed to parse (Missing texvc executable; please see math/README to configure.): x\preceq y

and Failed to parse (Missing texvc executable; please see math/README to configure.): y\preceq x
it is interpreted that the consumer is indifferent between Failed to parse (Missing texvc executable; please see math/README to configure.): x
and Failed to parse (Missing texvc executable; please see math/README to configure.): y
but is no reason to conclude that Failed to parse (Missing texvc executable; please see math/README to configure.): x=y

, preference relations are never assumed to be antisymmetric. In this context, for any Failed to parse (Missing texvc executable; please see math/README to configure.): B\subset X , we call Failed to parse (Missing texvc executable; please see math/README to configure.): x\in B

a maximal element if

Failed to parse (Missing texvc executable; please see math/README to configure.): y\in B

implies Failed to parse (Missing texvc executable; please see math/README to configure.):  y\preceq x

and it is interpreted as a consumption bundle that is not dominated by any other bundle in the sense that Failed to parse (Missing texvc executable; please see math/README to configure.): x\prec y , that is Failed to parse (Missing texvc executable; please see math/README to configure.): x\preceq y

and not Failed to parse (Missing texvc executable; please see math/README to configure.):  y\preceq x

.

It should be remarked that the formal definition looks very much like that of a greatest element for an ordered set. However, when Failed to parse (Missing texvc executable; please see math/README to configure.): \preceq

is only a preorder, an element Failed to parse (Missing texvc executable; please see math/README to configure.): x
with the property above behaves very much like a maximal element in an ordering. For instance, a maximal element Failed to parse (Missing texvc executable; please see math/README to configure.): x\in B
is not unique for Failed to parse (Missing texvc executable; please see math/README to configure.):  y\preceq x
does not preclude the possibility that Failed to parse (Missing texvc executable; please see math/README to configure.):  x\preceq y
(while Failed to parse (Missing texvc executable; please see math/README to configure.):  y\preceq x
and Failed to parse (Missing texvc executable; please see math/README to configure.):  x\preceq y
do not imply Failed to parse (Missing texvc executable; please see math/README to configure.):  x = y
but simply indifference Failed to parse (Missing texvc executable; please see math/README to configure.): x\sim y

). The notion of greatest element for a preference preorder would be that of most preferred choice. That is, some Failed to parse (Missing texvc executable; please see math/README to configure.): x\in B

with

Failed to parse (Missing texvc executable; please see math/README to configure.): y\in B

implies Failed to parse (Missing texvc executable; please see math/README to configure.):  y\prec x.

An obvious application is to the definition of demand correspondence. Let Failed to parse (Missing texvc executable; please see math/README to configure.): P

be the class of functionals on Failed to parse (Missing texvc executable; please see math/README to configure.): X

. An element Failed to parse (Missing texvc executable; please see math/README to configure.): p\in P

is called a price functional or price system and maps every consumption bundle Failed to parse (Missing texvc executable; please see math/README to configure.): x\in X
into its market value Failed to parse (Missing texvc executable; please see math/README to configure.): p(x)\in \Bbb{R}_+

. The budget correspondence is a correspondence Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma : P\times \Bbb{R}_+ \rightarrow X

mapping any price system and any level of income into a subset

Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma (p,m)=\{x\in X:p(x)\leq m\}.

The demand correspondence maps any price Failed to parse (Missing texvc executable; please see math/README to configure.): p

and any level of income Failed to parse (Missing texvc executable; please see math/README to configure.): m
into the set of Failed to parse (Missing texvc executable; please see math/README to configure.): \preceq 

-maximal elements of Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma (p,m) .

Failed to parse (Missing texvc executable; please see math/README to configure.): D(p,m)=\{x\in X:x

is a maximal element of Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma (p,m)\}

.

It is called demand correspondence because the theory predicts that for Failed to parse (Missing texvc executable; please see math/README to configure.): p

and Failed to parse (Missing texvc executable; please see math/README to configure.): m
given, the rational choice of a consumer Failed to parse (Missing texvc executable; please see math/README to configure.): x^*
will be some element Failed to parse (Missing texvc executable; please see math/README to configure.): x^*\in D(p,m)

.cs:Maximální a minimální prvek de:Maximales und minimales Element es:Elemento maximal fr:Élément maximal pl:Element minimalny i maksymalny ro:Maximal