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IEVref: | 102-01-09 | ID: | |

Language: | en | Status: Standard | |

Term: | order relation | ||

Synonym1: | order [Preferred] | ||

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Definition: | binary relation ℛ between elements a and b of a given set having the following properties: - reflexivity:
*a*ℛ*a***,** - antisymmetry: if
*a*ℛ*b*and*b*ℛ*a*then*a*=*b***,** - transitivity: if
*a*ℛ*b*and*b*ℛ*c*then*a*ℛ*c*, for any elements*a*,*b*and*c*of the given set
Note 1 to entry: The given set is said to be ordered by the relation ℛ. Note 2 to entry: An order relation is a total order if at least one of the relations Note 3 to entry: An order relation is a partial order if, for at least two elements | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO | ||

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- reflexivity:
*a*ℛ*a***,** - antisymmetry: if
*a*ℛ*b*and*b*ℛ*a*then*a*=*b***,** - transitivity: if
*a*ℛ*b*and*b*ℛ*c*then*a*ℛ*c*, for any elements*a*,*b*and*c*of the given set

Note 1 to entry: The given set is said to be ordered by the relation ℛ.

Note 2 to entry: An order relation is a total order if at least one of the relations *a*ℛ*b* and *b*ℛ*a* is true for any elements *a* and *b*. The usual order for real numbers is a total order because *a* ≤ *b* or *b* ≤ *a*.

Note 3 to entry: An order relation is a partial order if, for at least two elements *a* and *b*, neither *a*ℛ*b* nor *b*ℛ*a* is true. Examples are the divisibility relation for natural numbers and the inclusion relation for subsets of a set with at least two elements.