total order
A totally ordered set^{} (or linearly ordered set) is a poset $(T,\le )$ which has the property of comparability:

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for all $x,y\in T$, either $x\le y$ or $y\le x$.
In other words, a totally ordered set is a set $T$ with a binary relation^{} $\le $ on it such that the following hold for all $x,y,z\in T$:

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$x\le x$. (reflexivity^{})

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If $x\le y$ and $y\le x$, then $x=y$. (antisymmetry)

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If $x\le y$ and $y\le z$, then $x\le z$. (transitivity)

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Either $x\le y$ or $y\le x$. (comparability)
The binary relation $\le $ is then called a total order or a linear order (or total ordering or linear ordering). A totally ordered set is also sometimes called a chain, especially when it is considered as a subset of some other poset. If every nonempty subset of $T$ has a least element, then the total order is called a wellorder (http://planetmath.org/WellOrderedSet).
Some people prefer to define the binary relation $$ as a total order, rather than $\le $. In this case, $$ is required to be transitive^{} (http://planetmath.org/Transitive3) and to obey the law of trichotomy. It is straightforward to check that this is equivalent^{} to the above definition, with the usual relationship between $$ and $\le $ (that is, $x\le y$ if and only if either $$ or $x=y$).
A totally ordered set can also be defined as a lattice^{} $(T,\vee ,\wedge )$ in which the following property holds:

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for all $x,y\in T$, either $x\wedge y=x$ or $x\wedge y=y$.
Then totally ordered sets are distributive lattices (http://planetmath.org/DistributiveLattice).
Title  total order 
Canonical name  TotalOrder 
Date of creation  20130322 11:43:35 
Last modified on  20130322 11:43:35 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  25 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 06A05 
Classification  msc 91B12 
Classification  msc 5500 
Classification  msc 5501 
Synonym  linear order 
Synonym  total ordering 
Synonym  linear ordering 
Related topic  PartialOrder 
Related topic  Relation 
Related topic  SortingProblem 
Related topic  OrderedRing 
Related topic  ProofOfGeneralizedIntermediateValueTheorem 
Related topic  LinearContinuum 
Defines  totally ordered set 
Defines  linearly ordered set 
Defines  comparability 
Defines  totally ordered 
Defines  linearly ordered 
Defines  chain 
Defines  totallyordered set 
Defines  linearlyordered set 
Defines  totallyordered 
Defines  linearlyordered 