In

^{''n''}, each of these make it an ^{''n''} on that subset.

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a total or linear order is a partial order
upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, affirmative action to change the circumstances and habits that leads to s ...

in which any two elements are comparable. That is, a total order is a binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...

$\backslash leq$ on some set $X$, which satisfies the following for all $a,\; b$ and $c$ in $X$:
# $a\; \backslash leq\; a$ ( reflexive).
# If $a\; \backslash leq\; b$ and $b\; \backslash leq\; c$ then $a\; \backslash leq\; c$ (transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

)
# If $a\; \backslash leq\; b$ and $b\; \backslash leq\; a$ then $a\; =\; b$ ( antisymmetric)
# $a\; \backslash leq\; b$ or $b\; \backslash leq\; a$ (strongly connected Graph with strongly connected components marked
In the mathematical theory of directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a Graph (discrete mathematics), graph that is made up of a set of ...

, formerly called total).
Total orders are sometimes also called simple, connex, or full orders.
A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set.
An extension of a given partial order to a total order is called a linear extension
In order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, c ...

of that partial order.
Strict and non-strict total orders

A on a set $X$ is astrict partial order
In mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

on $X$ in which any two elements are comparable. That is, a total order is a binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...

$<$ on some set $X$, which satisfies the following for all $a,\; b$ and $c$ in $X$:
# Not $a\; <\; a$ (irreflexive
In mathematics, a homogeneous binary relation ''R'' over a set (mathematics), set ''X'' is reflexive if it relates every element of ''X'' to itself.
An example of a reflexive relation is the relation "equality (mathematics), is equal to" on the se ...

).
# If $a\; <\; b$ and $b\; <\; c$ then $a\; <\; c$ (transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

).
# If $a\; \backslash neq\; b$, then $a\; <\; b$ or $b\; <\; a$ ( connected).
For each (non-strict) total order $\backslash leq$ there is an associated relation $<$, called the ''strict total order'' associated with $\backslash leq$ that can be defined in two equivalent ways:
* $a\; <\; b$ if $a\; \backslash leq\; b$ and $a\; \backslash neq\; b$ ( reflexive reduction).
* $a\; <\; b$ if not $b\; \backslash leq\; a$ (i.e., $<$ is the complement
A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be:
* Complement (linguistics), a word or phrase having a particular syntactic role
** Subject complement, a word or phrase addi ...

of the converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a categorical or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical ...

of $\backslash leq$).
Conversely, the reflexive closureIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

of a strict total order $<$ is a (non-strict) total order.
Examples

* Anysubset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of a totally ordered set is totally ordered for the restriction of the order on .
* The unique order on the empty set, , is a total order.
* Any set of cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

s or ordinal number
In set theory
illustrating the intersection of two sets
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 s ...

s (more strongly, these are well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a Set (mathematics), 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. The set ''S'' together ...

s).
* If is any set and an injective function
In , an injective function (also known as injection, or one-to-one function) is a that maps elements to distinct elements; that is, implies . In other words, every element of the function's is the of one element of its . The term must no ...

from to a totally ordered set then induces a total ordering on by setting if and only if .
* The lexicographical order
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

on the Cartesian product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of a family of totally ordered sets, indexed
Index may refer to:
Arts, entertainment, and media Fictional entities
* Index (A Certain Magical Index), Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo (megastr ...

by a well ordered set, is itself a total order.
* The set of real numbers
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

ordered by the usual "less than or equal to" (≤) or "greater than or equal to" (≥) relations is totally ordered, and hence so are the subsets of natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

, integers
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

, and rational numbers
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. Each of these can be shown to be the unique (up to an order isomorphismIn the mathematical field of order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (g ...

) "initial example" of a totally ordered set with a certain property, (here, a total order is ''initial'' for a property, if, whenever has the property, there is an order isomorphism from to a subset of ):
** The natural numbers form an initial non-empty totally ordered set with no upper bound
In mathematics, particularly in order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

.
** The integers form an initial non-empty totally ordered set with neither an upper nor a lower bound
Lower may refer to:
* Lower (surname)
* Lower Township, New Jersey
*Lower Receiver (firearms)
* Lower Wick Gloucestershire, England
See also
* Nizhny
{{Disambiguation ...

.
** The rational numbers form an initial totally ordered set which is dense
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...

in the real numbers. Moreover, the reflexive reduction < is a dense orderIn mathematics, a partial order or total order < on a Set (mathematics), set $X$ is said to be dense if, for all $x$ and $y$ in $X$ for which $x\; <\; y$, there is a $z$ in $X$ ...

on the rational numbers.
** The real numbers form an initial unbounded totally ordered set that is connected in the order topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(defined below).
* Ordered fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s are totally ordered by definition. They include the rational numbers and the real numbers. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any ''Dedekind-complete
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

'' ordered field is isomorphic to the real numbers.
* The letters of the alphabet ordered by the standard dictionary order, e.g., etc., is a strict total order.
Chains

The term chain is sometimes defined as a synonym for a totally ordered set, but it is generally used for referring to asubset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of a partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

that is totally ordered for the induced order. Typically, the partially ordered set is a set of subsets of a given set that is ordered by inclusion, and the term is used for stating properties of the set of the chains. This high number of nested levels of sets explains the usefulness of the term.
A common example of the use of ''chain'' for referring to totally ordered subsets is Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...

which asserts that, if every chain in a partially ordered set has an upper bound in , then contains at least one maximal element. Zorn's lemma is commonly used with being a set of subsets; in this case, the upperbound is obtained by proving that the union of the elements of a chain in is in . This is the way that is generally used to prove that a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

has Hamel bases
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

and that a ring has maximal idealIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s.
In some contexts, the chains that are considered are order isomorphic to the natural numbers with their usual order or its opposite order. In this case, a chain can be identified with a monotone sequence, and is called an ascending chain or a descending chain, depending whether the sequence is increasing or decreasing.
A partially ordered set has the descending chain conditionIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

if every descending chain eventually stabilizes. For example, an order is well founded if it has the descending chain condition. Similarly, the ascending chain conditionIn mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings.Jacobson (2009), p. 1 ...

means that every ascending chain eventually stabilizes. For example, a Noetherian ring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

is a ring whose ideals
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...

satisfy the ascending chain condition.
In other contexts, only chains that are finite set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s are considered. In this case, one talks of a ''finite chain'', often shortened as a ''chain''. In this case, the length of a chain is the number of inequalities (or set inclusions) between consecutive elements of the chain; that is, the number minus one of elements in the chain. Thus a singleton set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a chain of length zero, and an ordered pair
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

is a chain of length one. The dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

of a space is often defined or characterized as the maximal length of chains of subspaces. For example, the dimension of a vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

is the maximal length of chains of linear subspace
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s, and the Krull dimension
In commutative algebra
Commutative algebra is the branch of algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with ...

of a commutative ring
In ring theory
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...

is the maximal length of chains of prime ideal
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

s.
"Chain" may also be used for some totally ordered subsets of structures
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A sy ...

that are not partially ordered sets. An example is given by regular chainIn computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of Wu's method of characteristic set, characteristic set.
Introduction
Given a System of linear equ ...

s of polynomials. Another example is the use of "chain" as a synonym for a walk
Walking (also known as ambulation) is one of the main gait
Gait is the pattern of movement
Movement may refer to:
Common uses
* Movement (clockwork), the internal mechanism of a timepiece
* Motion (physics), commonly referred to as move ...

in a graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

.
Further concepts

Lattice theory

One may define a totally ordered set as a particular kind of lattice, namely one in which we have : $\backslash \; =\; \backslash $ for all ''a'', ''b''. We then write ''a'' ≤ ''b''if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

$a\; =\; a\backslash wedge\; b$. Hence a totally ordered set is a distributive lattice
In mathematics, a distributive lattice is a lattice (order), lattice in which the operations of join and meet distributivity, distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice ope ...

.
Finite total orders

A simplecounting
Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...

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. Either by direct proof or by observing that every well order is order isomorphicIn the mathematical field of order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (g ...

to an ordinal
Ordinal may refer to:
* Ordinal data, a statistical data type consisting of numerical scores that exist on an arbitrary numerical scale
* Ordinal date, a simple form of expressing a date using only the year and the day number within that year
* O ...

one may show that every finite total order is order isomorphicIn the mathematical field of order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (g ...

to an initial segment
A Hasse diagram of the power set of the set with the upper set ↑ colored green. The white sets form the lower set ↓.
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered s ...

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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).
Category theory

Totally ordered sets form afull subcategory
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

of partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

s, with the morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s being maps which respect the orders, i.e. maps ''f'' such that if ''a'' ≤ ''b'' then ''f''(''a'') ≤ ''f''(''b'').
A bijective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

map
A map is a symbol
A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

between two totally ordered sets that respects the two orders is an isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

in this category.
Order topology

For any totally ordered set ''X'' we can define the ''open interval
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s'' (''a'', ''b'') = , (−∞, ''b'') = , (''a'', ∞) = and (−∞, ∞) = ''X''. We can use these open intervals to define a topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

on any ordered set, the order topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
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 .
Completeness

A totally ordered set is said to be complete if every nonempty subset that has anupper bound
In mathematics, particularly in order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

, has a least upper bound
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. For example, the set of real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s R is complete but the set of rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s Q is not. In other words, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s a property of the relation ≤ is that every non-empty #REDIRECT Empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

subset ''S'' of R with an upper bound
In mathematics, particularly in order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

in R has a least upper bound
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

(also called supremum) in R. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ≤ to the rational numbers.
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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

is compact
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...

. Examples are the closed intervals of real numbers, e.g. the unit interval
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

,1 and the affinely extended real number system
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and where the infinities are treated as actual numbers. It is useful in describing the algebra on infiniti ...

(extended real number line). There are order-preserving homeomorphism
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...

s between these examples.
Sums of orders

For any two disjoint total orders $(A\_1,\backslash le\_1)$ and $(A\_2,\backslash le\_2)$, there is a natural order $\backslash le\_+$ on the set $A\_1\backslash cup\; A\_2$, which is called the sum of the two orders or sometimes just $A\_1+A\_2$: : For $x,y\backslash in\; A\_1\backslash cup\; A\_2$, $x\backslash le\_+\; y$ holds if and only if one of the following holds: :# $x,y\backslash in\; A\_1$ and $x\backslash le\_1\; y$ :# $x,y\backslash in\; A\_2$ and $x\backslash le\_2\; y$ :# $x\backslash in\; A\_1$ and $y\backslash in\; A\_2$ Intuitively, this means that the elements of the second set are added on top of the elements of the first set. More generally, if $(I,\backslash le)$ is a totally ordered index set, and for each $i\backslash in\; I$ the structure $(A\_i,\backslash le\_i)$ is a linear order, where the sets $A\_i$ are pairwise disjoint, then the natural total order on $\backslash bigcup\_i\; A\_i$ is defined by : For $x,y\backslash in\; \backslash bigcup\_\; A\_i$, $x\backslash le\; y$ holds if: :# Either there is some $i\backslash in\; I$ with $x\backslash le\_i\; y$ :# or there are some $imath>\; in$ I$with$ x\backslash in\; A\_i$,$ y\backslash in\; A\_j$$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 theCartesian product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of two totally ordered sets are:
* Lexicographical order
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

: (''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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

). 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 productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

Rordered vector space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.
Definition
Given a vector space ''X'' over the real numbers R and a preo ...

.
See also examples of partially ordered sets.
A real function of ''n'' real variables defined on a subset of RRelated structures

A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is apartial order
upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, affirmative action to change the circumstances and habits that leads to s ...

.
A group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

with a compatible total order is a totally ordered group.
There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation. Forgetting the location of the ends results in a cyclic order
In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Inst ...

. Forgetting both data results in a separation relation.
See also

* Artinian ring * Order theory * Well-order * Suslin's problem * Countryman line * Permutation * Prefix order – a downward total partial orderNotes

References

* * * * George Grätzer (1971). ''Lattice theory: first concepts and distributive lattices.'' W. H. Freeman and Co. * John G. Hocking and Gail S. Young (1961). ''Topology.'' Corrected reprint, Dover, 1988. * *External links

* {{SpringerEOM , title=Totally ordered set , id=Total_order&oldid=35332 Binary relations Order theory Set theory