law of trichotomy

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

•

for all $x,y\in S$ , exactly one of the following holds: $x R y$ or $y R x$ 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	2013-03-22 14:13:46
Last modified on	2013-03-22 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