In
mathematics, especially in
order theory, a maximal element of a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
''S'' of some
preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defined
dually as an element of ''S'' that is not greater than any other element in ''S''.
The notions of maximal and minimal elements are weaker than those of
greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset
of a preordered set is an element of
which is greater than or equal to any other element of
and the minimum of
is again defined dually. In the particular case of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...
, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to
totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.
As an example, in the collection
ordered by
containment
Containment was a geopolitical strategic foreign policy pursued by the United States during the Cold War to prevent the spread of communism after the end of World War II. The name was loosely related to the term '' cordon sanitaire'', which w ...
, the element is minimal as it contains no sets in the collection, the element is maximal as there are no sets in the collection which contain it, the element is neither, and the element is both minimal and maximal. By contrast, neither a maximum nor a minimum exists for
Zorn's lemma states that every partially ordered set for which every totally ordered subset has an
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an elem ...
contains at least one maximal element. This lemma is equivalent to the
well-ordering theorem and the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
and implies major results in other mathematical areas like the
Hahn–Banach theorem, the
Kirszbraun theorem,
Tychonoff's theorem, the existence of a
Hamel basis for every vector space, and the existence of an
algebraic closure for every
field.
Definition
Let
be a
preordered set and let
is an element
such that
:if
satisfies
then necessarily
Similarly, is an element
such that
:if
satisfies
then necessarily
Equivalently,
is a minimal element of
with respect to
if and only if
is a maximal element of
with respect to
where by definition,
if and only if
(for all
).
If the subset
is not specified then it should be assumed that
Explicitly, a (respectively, ) is a maximal (resp. minimal) element of
with respect to
If the preordered set
also happens to be a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...
(or more generally, if the restriction
is a partially ordered set) then
is a maximal element of
if and only if
contains no element strictly greater than
explicitly, this means that there does not exist any element
such that
and
The characterization for minimal elements is obtained by using
in place of
Existence and uniqueness
Maximal elements need not exist.
*Example 1: Let
where
\R denotes the real numbers. For all
m \in S, s = m + 1 \in S but
m < s (that is,
m \leq s but not
m = s).
*Example 2: Let
S = \, where
\Q denotes the rational numbers and where Square_root_of_2#Proofs_of_irrationality,
\sqrt is irrational.
In general
\,\leq\, is only a partial order on
S. If
m is a maximal element and
s \in S, then it remains possible that neither
s \leq m nor
m \leq s. This leaves open the possibility that there exist more than one maximal elements.
*Example 3: In the
fence
A fence is a structure that encloses an area, typically outdoors, and is usually constructed from posts that are connected by boards, wire, rails or netting. A fence differs from a wall in not having a solid foundation along its whole length.
...
a_1 < b_1 > a_2 < b_2 > a_3 < b_3 > \ldots, all the
a_i are minimal and all
b_i are maximal, as shown in the image.
*Example 4: Let ''A'' be a set with at least two elements and let
S = \ be the subset of the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is ...
\wp(A) consisting of
singleton subsets, partially ordered by
\,\subseteq. This is the discrete poset where no two elements are comparable and thus every element
\ \in S is maximal (and minimal); moreover, for any distinct
a, b \in A, neither
\ \subseteq \ nor
\ \subseteq \.
Greatest elements
For a partially ordered set
(P, \leq), the
irreflexive kernel of
\,\leq\, is denoted as
\,<\, and is defined by
x < y if
x \leq y and
x \neq y.
For arbitrary members
x, y \in P, exactly one of the following cases applies:
#
x < y;
#
x = y;
#
y < x;
#
x and
y are incomparable.
Given a subset
S \subseteq P and some
x \in S,
* if case 1 never applies for any
y \in S, then
x is a maximal element of
S, as defined above;
* if case 1 and 4 never applies for any
y \in S, then
x is called a of
S.
Thus the definition of a greatest element is stronger than that of a maximal element.
Equivalently, a greatest element of a subset
S can be defined as an element of
S that is greater than every other element of
S.
A subset may have at most one greatest element.
[If g_1 and g_2 are both greatest, then g_1 \leq g_2 and g_2 \leq g_1, and hence g_1 = g_2 by antisymmetry. \blacksquare]
The greatest element of
S, if it exists, is also a maximal element of
S,[If g is the greatest element of S and s \in S, then s \leq g. By antisymmetry, this renders (g \leq s and g \neq s) impossible. \blacksquare] and the only one.
[If m is a maximal element then m \leq g (because g is greatest) and thus m = g since m is maximal. \blacksquare]
By
contraposition, if
S has several maximal elements, it cannot have a greatest element; see example 3.
If
P satisfies the
ascending chain condition, a subset
S of
P has a greatest element
if, and only if, it has one maximal element.
[: see above. — : Assume for contradiction that S has just one maximal element, m, but no greatest element. Since m is not greatest, some s_1 \in S must exist that is incomparable to m. Hence s_1 \in S cannot be maximal, that is, s_1 < s_2 must hold for some s_2 \in S. The latter must be incomparable to m, too, since m < s_2 contradicts m's maximality while s_2 \leq m contradicts the incomparability of m and s_1. Repeating this argument, an infinite ascending chain s_1 < s_2 < \ldots < s_n < \cdots can be found (such that each s_i is incomparable to m and not maximal). This contradicts the ascending chain condition. \blacksquare]
When the restriction of
\,\leq\, to
S is a
total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexiv ...
(
S = \ in the topmost picture is an example), then the notions of maximal element and greatest element coincide.
[Let m \in S be a maximal element, for any s \in S either s \leq m or m \leq s. In the second case, the definition of maximal element requires that s = m, so it follows that s \leq m. In other words, m is a greatest element. \blacksquare]
This is not a necessary condition: whenever
S has a greatest element, the notions coincide, too, as stated above.
If the notions of maximal element and greatest element coincide on every two-element subset
S of
P. then
\,\leq\, is a total order on
P.[If a, b \in P were incomparable, then S = \ would have two maximal, but no greatest element, contradicting the coincidence. \blacksquare]
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
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
where only total orders are considered. This observation applies not only to totally ordered subsets of any partially ordered set, but also to their order theoretic generalization via
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
s. In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. If a directed set has a maximal element, it is also its greatest element,
[Let m \in D be maximal. Let x \in D be arbitrary. Then the common upper bound u of m and x satisfies u \ge m, so u=m by maximality. Since x\le u holds by definition of u, we have x\le m. Hence m is the greatest element. \blacksquare] and hence its only maximal element. For a directed set without maximal or greatest elements, see examples 1 and 2
above.
Similar conclusions are true for minimal elements.
Further introductory information is found in the article on
order theory.
Properties
* Each finite nonempty subset
S has both maximal and minimal elements. An infinite subset need not have any of them, for example, the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s
\Z with the usual order.
* The set of maximal elements of a subset
S is always an
antichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
The size of the largest antichain in a partially ordered set is known as its w ...
, that is, no two different maximal elements of
S are comparable. The same applies to minimal elements.
Examples
* In
Pareto efficiency, a Pareto optimum is a maximal element with respect to the partial order of Pareto improvement, and the set of maximal elements is called the Pareto frontier.
* In
decision theory, an
admissible decision rule is a maximal element with respect to the partial order of ''
dominating decision rule.''
* In
modern portfolio theory, the set of maximal elements with respect to the
product order on risk and return is called the
efficient frontier.
* In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, a set is
finite if and only if every non-empty
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s has a minimal element when ordered by the
inclusion relation.
* In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, the concept of a
maximal common divisor is needed to generalize
greatest common divisors to number systems in which the common divisors of a set of elements may have more than one maximal element.
* In
computational geometry, the
maxima of a point set
In computational geometry, a point in a finite set of points is said to be ''maximal'' or ''non-dominated'' if there is no other point in whose coordinates are all greater than or equal to the corresponding coordinates of . The maxima of a poi ...
are maximal with respect to the partial order of coordinatewise domination.
Consumer theory
In economics, one may relax the axiom of antisymmetry, using preorders (generally
total preorders) instead of partial orders; the notion analogous to maximal element is very similar, but different terminology is used, as detailed below.
In
consumer theory the consumption space is some set
X, usually the positive orthant of some vector space so that each
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 \preceq so that
x, y \in X and
x \preceq y reads:
x is at most as preferred as
y. When
x\preceq y and
y \preceq x it is interpreted that the consumer is indifferent between
x and
y but is no reason to conclude that
x = y. preference relations are never assumed to be antisymmetric. In this context, for any
B \subseteq X, an element
x \in B is said to be a maximal element if
y \in B implies
y \preceq x
where it is interpreted as a consumption bundle that is not dominated by any other bundle in the sense that
x \prec y, that is
x \preceq y and not
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
\preceq is only a preorder, an element
x with the property above behaves very much like a maximal element in an ordering. For instance, a maximal element
x \in B is not unique for
y \preceq x does not preclude the possibility that
x \preceq y (while
y \preceq x and
x \preceq y do not imply
x = y but simply indifference
x \sim y). The notion of greatest element for a preference preorder would be that of most preferred choice. That is, some
x\in B with
y \in B implies
y \prec x.
An obvious application is to the definition of demand correspondence. Let
P be the class of functionals on
X. An element
p\in P is called a price functional or price system and maps every consumption bundle
x\in X into its market value
p(x)\in \R_+. The budget correspondence is a correspondence
\Gamma \colon P \times \R_ \rightarrow X mapping any price system and any level of income into a subset
\Gamma (p,m) = \.
The demand correspondence maps any price
p and any level of income
m into the set of
\preceq -maximal elements of
\Gamma (p, m).
D(p,m) = \left\.
It is called demand correspondence because the theory predicts that for
p and
m given, the
rational choice of a consumer
x^* will be some element
x^* \in D(p,m).
Related notions
A subset
Q of a partially ordered set
P is said to be
cofinal if for every
x \in P there exists some
y \in Q such that
x \leq y. Every cofinal subset of a partially ordered set with maximal elements must contain all maximal elements.
A subset
L of a partially ordered set
P is said to be a
lower set of
P if it is downward closed: if
y \in L and
x \leq y then
x \in L. Every lower set
L of a finite ordered set
P is equal to the smallest lower set containing all maximal elements of
L.
See also
*
*
*
Notes
;Proofs
References
{{DEFAULTSORT:Maximal Element
Order theory