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# betweenness relation

# 1 Definition

Let $A$ be a set. A ternary relation $B$ on
$A$ is said to be a *betweenness relation* if it has the following properties:

- O1
if $(a,b,c)\in B$, then $(c,b,a)\in B$; in other words, the set

$B(b)=\{(a,c)\mid(a,b,c)\in B\}$ is a symmetric relation for

*each*$b$; thus, from now on, we may say, without any ambiguity, that $b$ is*between*$a$ and $c$ if $(a,b,c)\in B$; - O2
if $(a,b,a)\in B$, then $a=b$;

- O3
for each $a,b\in A$, there is a $c\in A$ such that $(a,b,c)\in B$;

- O4
for each $a,b\in A$, there is a $c\in A$ such that $(a,c,b)\in B$;

- O5
if $(a,b,c)\in B$ and $(b,a,c)\in B$, then $a=b$;

- O6
if $(a,b,c)\in B$ and $(b,c,d)\in B$, then $(a,b,d)\in B$;

- O7
if $(a,b,d)\in B$ and $(b,c,d)\in B$, then $(a,b,c)\in B$.

Related:

SomeTheoremsOnTheAxiomsOfOrder

Synonym:

axioms of order

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

51G05*no label found*

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