In set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, a total order, linear order, simple order, or (non-strict) ordering is a binary relation In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations (here denoted by infix Infix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on . It is not as simple to parse by computers as prefix notation ( e.g. + 2 2 ) or postfix notation ( e.g. 2 2 + ), but many programming languages use it due to its familiarity ≤) on some set A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In X. The relation is transitive In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c, antisymmetric Partial and total orders are antisymmetric by definition. Therefore the usual order relation ≤ on the real numbers, the subset order ⊆ on the subsets of any given set and the divisibility order of the natural numbers are antisymmetric. For example, if for two real numbers x and y both inequalities x ≤ y and y ≤ x hold then x and y must be, and total In mathematics, a binary relation R over a set X is total if for all a and b in X, a is related to b or b is related to a. A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain.

If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:

If a ≤ b and b ≤ a then a = b (antisymmetry Partial and total orders are antisymmetric by definition. Therefore the usual order relation ≤ on the real numbers, the subset order ⊆ on the subsets of any given set and the divisibility order of the natural numbers are antisymmetric. For example, if for two real numbers x and y both inequalities x ≤ y and y ≤ x hold then x and y must be);

If a ≤ b and b ≤ c then a ≤ c (transitivity In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c);

a ≤ b or In logic and mathematics, or, also known as logical disjunction or inclusive disjunction, is a logical operator that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are true. In grammar, or is a coordinating conjunction. In ordinary b ≤ a (totality In mathematics, a binary relation R over a set X is total if for all a and b in X, a is related to b or b is related to a).

Contrast with a partial order In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. These relations are, which has a weaker form of the third condition (it only requires reflexivity In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation R on S where xRx holds true for every x in S, not totality). A relation having the property of "totality" means that any pair of elements in the set of the relation are mutually comparable under the relation.

Totality implies reflexivity, that is, a ≤ a. Thus a total order is also a partial order In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. These relations are, that is, a binary relation In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations which is reflexive, antisymmetric and transitive. Hence a total order is also a partial order satisfying the "totality" condition.

Strict total order

For each (non-strict) total order ≤ there is an associated asymmetric Asymmetric often means, simply: not symmetric. In this sense an asymmetric relation is a binary relation which is not a symmetric relation (hence irreflexive) relation <, called a strict total order, which can equivalently be defined in two ways:

• a < b if and only if In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional ("if") combined with its reverse ("only if"); hence the name. The result is that the truth of a ≤ b and a ≠ b
• a < b if and only if not b ≤ a (i.e., < is the inverse In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations of the complement In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations of ≤)

Properties:

• The relation is transitive: a < b and b < c implies a < c.
• The relation is trichotomous Generally, a trichotomy is a splitting into three disjoint parts. In mathematics, the law of trichotomy is most commonly the statement that for any (real) numbers x and y, exactly one of the following relations holds:: exactly one of a < b, b < a and a = b is true.
• The relation is a strict weak order, where the associated equivalence is equality.

We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤ can equivalently be defined in two ways:

• a ≤ b if and only if a < b or a = b
• a ≤ b if and only if not b < a

Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.

We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whether we are talking about the non-strict or the strict total order.

Examples

• The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc.

• Any subset of a totally ordered set, with the restriction of the order on the whole set.

• Any set of cardinal numbers In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets or ordinal numbers In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated (more strongly, these are well-orders In mathematics, a well-order relation on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded total order. The set S together with the well-order relation is then called a well-ordered set).

• If X is any set and f an injective function In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of its codomain is mapped to by at most one element of its domain. If in addition all of the elements in the codomain are in fact mapped to by some element from X to a totally ordered set then f induces a total ordering on X by setting x₁ < x₂ if and only if f(x₁) < f(x₂).

• The lexicographical order In mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic product), is a natural order structure of the Cartesian product of two ordered sets on the Cartesian product In mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept of a set of totally ordered sets indexed by an ordinal, is itself a total order. For example, any set of words ordered alphabetically is a totally ordered set, viewed as a subset of a Cartesian product of a countable number of copies of a set formed by adding the space symbol to the alphabet (and defining a space to be less than any letter).

• The set of real numbers In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an ordered by the usual less than (<) or greater than (>) relations is totally ordered, hence also the subsets of natural numbers In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers{0, 1, 2, ...} according to a definition first appearing in the nineteenth century, integers The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 6, and rational numbers In mathematics a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q , which stands for quotient. Each of these can be shown to be the unique (to within isomorphism) smallest example of a totally ordered set with a certain property, (a total order A is the smallest with a certain property if whenever B has the property, there is an order isomorphism from A to a subset of B):
  • The natural numbers comprise the smallest totally ordered set with no upper bound In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S. A set with an upper bound is said to be bounded from above by.
  • The integers comprise the smallest totally ordered set with neither an upper nor a lower bound In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S. A set with an upper bound is said to be bounded from above by.
  • The rational numbers comprise the smallest totally ordered set with no upper or lower bound, which is dense in the sense that for every a and b such that a < b there is a c such that a < c < b.
  • The real numbers comprise the smallest unbounded connected In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component totally ordered set. (See below for the definition of the topology.)

Further concepts

Chains

While chain is sometimes merely a synonym for totally ordered set, it can also refer to a totally ordered subset In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B of some partially ordered set In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. These relations are. The latter definition has a crucial role in Zorn's lemma Every partially ordered set in which every chain has an upper bound contains at least one maximal element.

For example, consider the set of all subsets of the integers The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 6 partially ordered In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. These relations are by inclusion In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B. Then the set { I_n : n is a natural number In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century}, where I_n is the set of natural numbers below n, is a chain in this ordering, as it is totally ordered under inclusion: If n≤k, then I_n is a subset of I_k.

Lattice theory

One may define a totally ordered set as a particular kind of lattice In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum (the elements' least upper bound; called their join) and an infimum (greatest lower bound; called their meet). Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are, namely one in which we have

for all a, b.

We then write a ≤ b if and only if In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional ("if") combined with its reverse ("only if"); hence the name. The result is that the truth of . Hence a totally ordered set is a distributive lattice; here is the proof.

Finite total orders

A simple counting Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; argument will verify that any non-empty finite totally-ordered set (and hence any non-empty subset thereof) has a least element. Thus every finite total order is in fact a well order In mathematics, a well-order relation on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded total order. The set S together with the well-order relation is then called a well-ordered set. Either by direct proof or by observing that every well order is order isomorphic to an ordinal In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated one may show that every finite total order is order isomorphic to an initial segment of the natural numbers ordered by <. In other words a total order on a set with k elements induces a bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with order type In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection f: X → Y such that both f and its inverse are monotone . (In the special case when X is totally ordered, monotonicity of f implies monotonicity of its inverse.) ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).

Category theory

Totally ordered sets form a full subcategory In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and arrows of the category In mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may be abstract entities of any kind of partially ordered sets In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. These relations are, with the morphisms In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures being maps which respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b).

A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.

Order topology

For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, the order topology.

When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology on N induced by < and the order topology on N induced by > (in this case they happen to be identical but will not in general).

The order topology induced by a total order may be shown to be hereditarily normal.

Completeness

A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upper bound. For example, the set of real numbers R is complete but the set of rational numbers Q is not.

There are a number of results relating properties of the order topology to the completeness of X:

• If the order topology on X is connected, X is complete.
• X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is two points a and b in X with a < b such that no c satisfies a < c < b.)
• X is complete if and only if every bounded set that is closed in the order topology is compact.

A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervals of real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line). There are order-preserving homeomorphisms between these examples.

Orders on the Cartesian product of totally ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product of two totally ordered sets are:

• Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order.
• (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order.
• (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of the corresponding strict total orders). This is also a partial order.

All three can similarly be defined for the Cartesian product of more than two sets.

Applied to the vector space Rⁿ, each of these make it an ordered vector space.

See also examples of partially ordered sets.

A real function of n real variables defined on a subset of Rⁿ defines a strict weak order and a corresponding total preorder on that subset.

See also

• Linear extension
• Order theory
• Well-order
• Suslin's problem
• Countryman line

Notes

References

• George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN 0-7167-0442-0
• John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4

Categories: Mathematical relations | Order theory | Set theory

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