Consider a po-group $(G,\cdot,1,\leq)$ as a small category, a convex normal subgroup $S\vartriangleleft G$ and the group actions of $G$ on $\leq$ via left and right operations. The orbits are of the form
$$ S(x,y)S := \{ (gxh,gyh) ∈ {\leq} \mid g,h∈S\}$$
Is there any such setting that for given elements $x,y∈G$ there exists an Element $z∈G$ with $x\leq z \leq y$ such that the following does not hold:
$$ S(x,z)S*S(z,y)S:= \{ (a,c) ∈ {\leq} \mid ∃b∈G:(a,b) ∈ S(x,z)S, (b,c)∈ S(z,y)S\} \subseteq S(x,y)S?$$
Is there some nice counter example?
Actually it is sufficient to consider the orbit of $(z,y)$ of the stabilizer of $z$ under the combined action of the multiplication from left and from right. This could be from (in block operations) $$ S(x,z)S*S(z,y)S = S\bigl((x,z)*S(z,y)S\bigr)S = S\bigl((x,z)*(z,y)(S^d\times S)_z\bigr)S,$$
Where $S^d\times S$ is the combined group (action) of the multiplication from left and from right.
Edit: Fixed the usage of the groups (all $S$ were acciently replaced by $G$).