Definition

Let

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
$\displaystyle B(b)= \lbrace (a,c)\mid (a,b,c)\in B\rbrace$
is a symmetric relation for each ; thus, from now on, we may say, without any ambiguity, that is between and if $(a,b,c)\in B$ ;
O2: if $(a,b,a)\in B$ , then ;
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 ;
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 $(a,b,c)\in B$ .

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Other names:

axioms of order

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Cross-references: properties, ternary relation
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This is version 3 of betweenness relation, born on 2007-06-24, modified 2007-06-24.
Object id is 9661, canonical name is BetweennessRelation.
Accessed 816 times total.

Classification:

51G05 (Geometry :: Ordered geometries )

Pending Errata and Addenda

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