linear ordered geometry

An incidence geometry $A=(P,n,I)$ is a linear ordered geometry if there is a strict betweenness relation $B$ defined on the points $P_{0}$ of $A$ , such that

Col1

$(p,q,r)\in B$ only if $p, q$ , and $r$ are collinear (all incident with a common line $\ell\in P_{1}$ );
Col2

for any pairwise distinct collinear points $p, q, r$ , at least one of $(p,q,r)$ , $(q,r,p)$ , or $(r,p,q)\in B$ ,

We denote the linear ordered geometry by $(A,B)$ .