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