According to Wikipedia, a partial cyclic order is a ternary relation $R$ that is
- cyclic: $R(x,y,z) \Rightarrow R(y,z,x)$,
- asymmetric: $R(x,y,z) \Rightarrow \text{not }R(z,y,x)$,
- transitive: $R(x,y,z)$ and $R(x,z,u) \Rightarrow R(x,y,u)$.
Seeking to understand the intuition behind that definition, I first thought that $R(x,y,z)$ intuitively means that $y$ is "between" $x$ and $z$ when reading these elements on some cycle. But in this naive approach, if $y$ is between $x$ and $z$ and if $z$ is between $y$ and $u$, then both $y$ and $z$ are between $x$ and $u$, which leads to the following property:
- $R(x,y,z)$ and $R(y,z,u) \Rightarrow R(x,y,u)$ and $R(x, z, u)$.
Question. Does a partial cyclic order satisfies (4)? And if a ternary relation satisfies (1), (2) and (4), is it a partial cyclic order? More generally, what is the connection, if any, between partial cyclic orders and betweenness?
No, implication (4) is not necessarily satisfied, because $u$ might go "too far" beyond $y$ and $z$ and cross over $x$ again. For example, let $R$ be the cyclic order on the numbers $1<2<3<4$. We have $R(1,3,4)$ and $R(3,4,2)$ but not $R(1,3,2)$.