law of trichotomy
The law of trichotomy for a binary relation^{} $R$ on a set $S$ is the property that

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for all $x,y\in S$, exactly one of the following holds: $xRy$ or $yRx$ or $x=y$.
A binary relation satisfying the law of trichotomy is sometimes said to be trichotomous. Trichotomous binary relations are equivalent^{} to tournaments^{}, although the study of tournaments is usually restricted to the finite case.
A transitive^{} trichotomous binary relation is called a total order^{}, and is typically written $$.
The law of trichotomy for cardinal numbers^{} is equivalent (in ZF) to the axiom of choice^{} (http://planetmath.org/AxiomOfChoice).
Title  law of trichotomy 

Canonical name  LawOfTrichotomy 
Date of creation  20130322 14:13:46 
Last modified on  20130322 14:13:46 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  9 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 06A05 
Classification  msc 03E20 
Defines  trichotomy 
Defines  trichotomous 