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| In [[mathematics]], a '''linear order''', '''total order''', '''simple order''', or '''(non-strict) ordering''' is a [[binary relation]] (here denoted by [[Infix notation|infix]] '''≤''') on some [[Set (mathematics)|set]] ''X'' which is [[transitive relation|transitive]], [[antisymmetric relation|antisymmetric]], and [[total relation|total]]. A set paired with a total order is called a '''totally ordered set''', a '''linearly ordered set''', a '''simply ordered set''', or a '''chain'''.
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| If ''X'' is totally ordered under ≤, then the following statements hold for all ''a'', ''b'' and ''c'' in ''X'':
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| : If ''a'' ≤ ''b'' and ''b'' ≤ ''a'' then ''a'' = ''b'' ([[antisymmetric relation|antisymmetry]]);
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| : If ''a'' ≤ ''b'' and ''b'' ≤ ''c'' then ''a'' ≤ ''c'' ([[transitive relation|transitivity]]);
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| : ''a'' ≤ ''b'' [[logical disjunction|or]] ''b'' ≤ ''a'' ([[total relation|totality]]).
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| Antisymmetry eliminates uncertain cases when both ''a'' precedes ''b'' and ''b'' precedes ''a''.<ref>{{cite book |last=Nederpelt |first=Rob |title=Logical Reasoning: A First Course |publisher=King's College Publications|year=2004| series=Texts in Computing|volume=3|edition=3rd, Revised |page=325 |chapter=Chapter 20.2: Ordered Sets. Orderings|isbn=0-9543006-7-X}}</ref> A relation having the property of "totality" means that any pair of elements in the set of the relation are [[Comparability|comparable]] under the relation. This also means that the set can be diagrammed as a line of elements, giving it the name ''linear''.<ref>{{cite book |last=Nederpelt |first=Rob |title=Logical Reasoning: A First Course |publisher=King's College Publications|year=2004| series=Texts in Computing|volume=3|edition=3rd, Revisied |page=330 |chapter=Chapter 20.3: Ordered Sets. Linear orderings|isbn=0-9543006-7-X}}</ref> ''Totality'' also implies reflexivity, i.e., ''a'' ≤ ''a''. Therefore, a total order is also a [[partial order]]. The [[partial order]] has a weaker form of the third condition (it only requires [[reflexive relation|reflexivity]], not totality). An extension of a given partial order to a total order is called a [[linear extension]] of that partial order.
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| ==Strict total order==
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| For each (non-strict) total order ≤ there is an associated [[asymmetric relation|asymmetric]] (hence irreflexive) relation <, called a '''strict total order''', which can equivalently be defined in two ways:
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| *''a'' < ''b'' [[if and only if]] ''a'' ≤ ''b'' and ''a'' ≠ ''b''
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| *''a'' < ''b'' if and only if not ''b'' ≤ ''a'' (i.e., < is the [[Binary_relation#Operations_on_binary_relations|inverse]] of the [[Binary_relation#Complement|complement]] of ≤)
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| Properties:
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| *The relation is transitive: ''a'' < ''b'' and ''b'' < ''c'' implies ''a'' < ''c''.
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| *The relation is [[Trichotomy (mathematics)|trichotomous]]: exactly one of ''a'' < ''b'', ''b'' < ''a'' and ''a'' = ''b'' is true.
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| *The relation is a [[strict weak order]], where the associated equivalence is equality.
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| 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:
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| *''a'' ≤ ''b'' if and only if ''a'' < ''b'' or ''a'' = ''b''
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| *''a'' ≤ ''b'' if and only if not ''b'' < ''a''
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| Two more associated orders are the complements ≥ and >, completing the [[Tuple|quadruple]] {<, >, ≤, ≥}.
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| 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.
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| ==Examples==
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| * The letters of the alphabet ordered by the standard [[Alphabetical order|dictionary order]], e.g., ''A'' < ''B'' < ''C'' etc.
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| * Any subset of a totally ordered set, with the restriction of the order on the whole set.
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| * Any set of [[cardinal number]]s or [[ordinal number]]s (more strongly, these are [[well-order]]s).
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| * If ''X'' is any set and ''f'' an [[injective function]] from ''X'' to a totally ordered set then ''f'' induces a total ordering on ''X'' by setting ''x''<sub>1</sub> < ''x''<sub>2</sub> if and only if ''f''(''x''<sub>1</sub>) < ''f''(''x''<sub>2</sub>).
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| * The [[lexicographical order]] on the [[Cartesian product]] of a set of totally ordered sets [[Index set|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 set|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).
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| * The set of ''[[real numbers]]'' ordered by the usual less than (<) or greater than (>) relations is totally ordered, hence also the subsets of ''[[natural numbers]]'', ''[[integers]]'', and ''[[rational numbers]]''. 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''):
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| **The ''natural numbers'' comprise the smallest totally ordered set with no [[upper bound]].
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| **The ''integers'' comprise the smallest totally ordered set with neither an upper nor a [[lower bound]].
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| **The ''rational numbers'' comprise the smallest totally ordered set which is ''[[dense order|dense]]'' in the real numbers. The definition of density used here says that for every 'a' and 'b' in the real numbers such that 'a' < 'b', there is a 'q' in the rational numbers such that 'a' < 'q' < 'b'.
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| **The ''real numbers'' comprise the smallest unbounded totally ordered set that is [[connectedness|connected]] in the order topology (defined below).
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| * [[Ordered field]]s are totally ordered by definition. They comprise the [[rational numbers]] and the real numbers.
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| ==Further concepts==
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| ===Chains===
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| While '''chain''' is sometimes merely a synonym for totally ordered set, it can also refer to a totally ordered [[subset]] of some [[partially ordered set]]. The latter definition has a crucial role in [[Zorn's lemma]].
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| For example, consider the set of all subsets of the [[integer]]s [[partially ordered]] by [[Subset|inclusion]]. Then the set { ''I''<sub>''n''</sub> : ''n'' is a [[natural number]]}, where ''I''<sub>''n''</sub> 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''<sub>''n''</sub> is a subset of ''I''<sub>''k''</sub>.
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| ===Lattice theory===
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| One may define a totally ordered set as a particular kind of [[lattice (order)|lattice]], namely one in which we have
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| : <math>\{a\vee b, a\wedge b\} = \{a, b\}</math> for all ''a'', ''b''.
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| We then write ''a'' ≤ ''b'' [[if and only if]] <math>a = a\wedge b</math>. Hence a totally ordered set is a [[distributive lattice]].
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| ===Finite total orders===
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| A simple [[counting]] 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 isomorphic]] to an [[Ordinal number|ordinal]] 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]] ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).
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| ===Category theory===
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| Totally ordered sets form a [[subcategory|full subcategory]] of the [[category (mathematics)|category]] of [[partially ordered set]]s, with the [[morphism]]s being maps which respect the orders, i.e. maps f such that if ''a'' ≤ ''b'' then ''f(a)'' ≤ ''f(b)''.
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| A [[bijection|bijective]] [[map (mathematics)|map]] between two totally ordered sets that respects the two orders is an [[isomorphism]] in this category.
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| ===Order topology===
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| For any totally ordered set ''X'' we can define the '''[[interval (mathematics)|open interval]]s''' (''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]].
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| 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).
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| The order topology induced by a total order may be shown to be hereditarily [[Normal space|normal]].
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| ===Completeness===<!-- This section is linked from [[Completely distributive lattice]]. See [[WP:MOS#Section management]] -->
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| 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 number]]s '''R''' is complete but the set of [[rational number]]s '''Q''' is not.
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| There are a number of results relating properties of the order topology to the completeness of X:
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| *If the order topology on ''X'' is connected, ''X'' is complete.
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| *''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''.)
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| *''X'' is complete if and only if every bounded set that is closed in the order topology is compact.
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| A totally ordered set (with its order topology) which is a [[complete lattice]] is [[Compact space|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 [[homeomorphism]]s between these examples.
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| ===Sums of orders===<!-- This section is linked from [[Scattered_order]]. See [[WP:MOS#Section management]] -->
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| For any two disjoint total orders <math>(A_1,\le_1)</math> and <math>(A_2,\le_2)</math>, there is a natural order <math>\le_+</math> on the set <math>A_1\cup A_2</math>, which is called the sum of the two orders or sometimes just <math>A_1+A_2</math>:
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| : For <math>x,y\in A_1\cup A_2</math>, <math>x\le_+ y</math> holds if and only if one of the following holds:
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| :# <math>x,y\in A_1</math> and <math>x\le_1 y</math>
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| :# <math>x,y\in A_2</math> and <math>x\le_2 y</math>
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| :# <math>x\in A_1</math> and <math>y\in A_2</math>
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| Intutitively, this means that the elements of the second set are added on top of the elements of the first set.
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| More generally, if <math>(I,\le)</math> is a totally ordered index set, and for each <math>i\in I</math> the structure <math>(A_i,\le_i)</math> is a linear order, where the sets <math>A_i</math> are pairwise disjoint, then the natural total order on <math>\bigcup_i A_i</math> is defined by
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| : For <math>x,y\in \bigcup_{i\in I} A_i</math>, <math>x\le y</math> holds if:
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| :# Either there is some <math>i\in I</math> with <math> x\le_i y </math>
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| :# or there are some <math>i<j</math> in <math>I</math> with <math> x\in A_i</math>, <math> y\in A_j</math>
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| ==Orders on the Cartesian product of totally ordered sets==
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| 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:
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| *[[Lexicographical order]]: (''a'',''b'') ≤ (''c'',''d'') if and only if ''a'' < ''c'' or (''a'' = ''c'' and ''b'' ≤ ''d''). This is a total order.
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| *(''a'',''b'') ≤ (''c'',''d'') if and only if ''a'' ≤ ''c'' and ''b'' ≤ ''d'' (the [[product order]]). This is a partial order.
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| *(''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#Direct_product_of_binary_relations|direct product]] of the corresponding strict total orders). This is also a partial order.
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| All three can similarly be defined for the Cartesian product of more than two sets.
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| Applied to the [[vector space]] '''R'''<sup>''n''</sup>, each of these make it an [[ordered vector space]].
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| See also [[Partially_ordered_set#Examples|examples of partially ordered sets]].
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| A real function of ''n'' real variables defined on a subset of '''R'''<sup>''n''</sup> [[Strict_weak_ordering#Function|defines a strict weak order and a corresponding total preorder]] on that subset.
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| == Related structures==
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| A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a [[partial order]].
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| A [[group (mathematics)|group]] with a compatible total order is a [[totally ordered group]].
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| 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]]. Forgetting both data results in a [[separation relation]].<ref>{{Citation |last=Macpherson |first=H. Dugald |year=2011 |title=A survey of homogeneous structures |journal=Discrete Mathematics |doi=10.1016/j.disc.2011.01.024 |url=http://www.amsta.leeds.ac.uk/pure/staff/macpherson/homog_final2.pdf |accessdate=28 April 2011}}</ref>
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| == See also ==
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| *[[Order theory]]
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| *[[Well-order]]
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| *[[Suslin's problem]]
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| *[[Countryman line]]
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| ==Notes==
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| {{Reflist|2}}
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| == References ==
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| * George Grätzer (1971). ''Lattice theory: first concepts and distributive lattices.'' W. H. Freeman and Co. ISBN 0-7167-0442-0
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| * John G. Hocking and Gail S. Young (1961). ''Topology.'' Corrected reprint, Dover, 1988. ISBN 0-486-65676-4
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| [[Category:Mathematical relations]]
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| [[Category:Order theory]]
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| [[Category:Set theory]]
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| [[de:Lineare Ordnung]]
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