Posets uniquely determined by its compact saturated subsets

Inspired by Zhao and Xu's study on which a dcpo can be determined by its Scott closed subsets lattice, we further investigate whether a poset (or dcpo) P is able to be determined by the family Q(P) of its Scott compact saturated subsets, in the sense that the isomorphism between ( Q(P), ⊇) and ( Q(M), ⊇) implies the isomorphism between P and M for any poset (or dcpo) M, in such case, P is called Qσ-unique. Quasicontinuous domains are proved to be Qσ-unique posets and draw support from which, we provide a class of Qσ-unique dcpos. We also define a new kind of posets called KD and show that every co-sober KD poset is Qσ-unique. It even yields another kind of Qσ-unique dcpos. It is gratifying that weakly well-filtered co-sober posets are also Qσ-unique. At last, we distinguish among the conditions which make a poset (or dcpo) Qσ-unique from each other by some examples; meanwhile, it is confirmed that none of them except the property of being co-sober are necessary for a poset (or dcpo) to be Qσ-unique.