Maximal and minimal elements

Definition

Let (P, \leq) be a preordered set and let S \subseteq P. A maximal element of S with respect to ,\leq, is an element m \in S such that

:if s \in S satisfies m \leq s, then necessarily s \leq m.

Similarly, a minimal element of S with respect to ,\leq, is an element m \in S such that

:if s \in S satisfies s \leq m, then necessarily m \leq s.

Equivalently, m \in S is a minimal element of S with respect to ,\leq, if and only if m is a maximal element of S with respect to ,\geq,, where by definition, q \geq p if and only if p \leq q (for all p, q \in P).

If the subset S is not specified then it should be assumed that S := P. Explicitly, a maximal element (respectively, minimal element) of (P, \leq) is a maximal (resp. minimal) element of S := P with respect to ,\leq.

If the preordered set (P, \leq) also happens to be a partially ordered set (or more generally, if the restriction (S, \leq) is a partially ordered set) then m \in S is a maximal element of S if and only if S contains no element strictly greater than m; explicitly, this means that there does not exist any element s \in S such that m \leq s and m \neq s. The characterization for minimal elements is obtained by using ,\geq, in place of ,\leq.

Existence and uniqueness

Maximal elements need not exist.

• Example 1: Let S = 1, \infty) \subseteq \R where \R denotes the [real numbers. For all m \in S, s = m + 1 \in S but m (that is, m \leq s but not m = s).
• Example 2: Let S = { s \in \Q : 1 \leq s^2 \leq 2 }, where \Q denotes the rational numbers and where \sqrt{2} 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_1 a_2 a_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 = { { a } : a \in A } be the subset of the power set \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 { a } \in S is maximal (and minimal); moreover, for any distinct a, b \in A, neither { a } \subseteq { b } nor { b } \subseteq { a }.

Greatest and least elements

Main article: Greatest and least elements

For a partially ordered set (P, \leq), the irreflexive kernel of ,\leq, is denoted as , and is defined by x if x \leq y and x \neq y. For arbitrary members x, y \in P, exactly one of the following cases applies:

1. x ;
2. x = y;
3. y ;
4. 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 greatest element 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.Only if: see above. — If: 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 must hold for some s_2 \in S. The latter must be incomparable to m, too, since m 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 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 (S = { 1, 2, 4 } 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 = { a, b } would have two maximal, but no greatest element, contradicting the coincidence. \blacksquare

Dual to greatest is the notion of least element that relates to minimal in the same way as greatest to maximal.

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 applies not only to totally ordered subsets of any partially ordered set, but also to their order theoretic generalization via directed sets. 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 integers \Z with the usual order.
• The set of maximal elements of a subset S is always an antichain, 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, a set is finite if and only if every non-empty family of subsets has a minimal element when ordered by the inclusion relation.
• In abstract algebra, 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 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) = { x \in X : p(x) \leq 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{ x \in X : x \text{ is a maximal element of } \Gamma(p,m) \right}.

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.

Notes

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References

References

1. (2009). "A Discrete Transition to Advanced Mathematics". American Mathematical Society.
2. Scott, William Raymond. (1987). "Group Theory". Dover.