I am looking for a reference for the following result:
Let $(X,\leq)$ be a linear order and let $\Pi$ be a partition of $X$ into convex subsets.
Then the relation $x\preceq y$ defined by $\forall P,Q\in \Pi,(x\in P\wedge y\in Q \rightarrow (\exists (p,q)\in P\times Q,(p\leq q)))$ is a quasi order. Moreover, for $x,y \in X$ we have $\exists P \in \Pi,x,y \in P$ if and only if $x\preceq y\wedge y \preceq x$ holds.
I thought I would find this in the book Linear Orderings by Rosenstein, but it does not occur; however it should exist somewhere...
Let P(x) be the unique A in P = $\Pi$ with x in A.
Easier to use is the equivalent definition
x <=' y when exists a in P(x), b in P(y) with a <= b.
Clearly, x <=' x.
If P(x) = P(y), then exists a in P(x) $\cap$ P(y) with a <= a.
. . Thus x <=' y and y <=' x.
If x <=' y and y <=' x, then exists
. . a,b in P(x), c,d in P(y) with a <= c, d <= b.
If d <= a, then a in P(y) and P(x) = P(y).
If a <= d, then d in P(x) and P(x) = P(y).
If x <=' y, y <=' z, then exists
. . a in P(x), b,c in P(y), d in P(z) with a <= b, c <= d.
If a <= c, then a <= d and x <=' z.
If c <= a, then a in P(y), so P(x) = P(y), c in P(x), x <=' z.
I prefer the alternative of ordering the partition P by
A <=* B when exists a in A, b in B with a <= b.
Clearly A <=* A.
If A <=* B, B <=* A, then
. . exists a,b in A, x,y in B with a <= x, y <= b.
If y <= a, then a in B, A = B.
If a <= y, then y in A, A = B.
If A <=* B, B <=* C, then
. . exists a in A, b,b' in B, c in C with a <= b, b' <= c.
If a <= b', then a <= c and A <=* C.
If b' <= a, then a in B, so A = B, A <=* C.
With that order for P, the quasi-order <=' can be recovered:
. . x <=' y iff exists A,B in P with x in A, y in B, A <=* B.