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betweenness relation (Definition)

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
$\displaystyle B(b)= \lbrace (a,c)\mid (a,b,c)\in B\rbrace$
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$.



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See Also: some theorems on the axioms of order

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:
AMS MSC51G05 (Geometry :: Ordered geometries )

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