$C_{\sigma }$-Unique Dcpos: Intrinsic Characterizations and Counterexamples

A dcpo $D$ is called $C_{\sigma }$-unique if for every dcpo $Q$, $\mathrm{\Gamma }(D)\cong \mathrm{\Gamma }(Q)$ implies $D\cong Q,$ where $\mathrm{\Gamma }(D)$ denotes the lattice of Scott-closed subsets of $D$. We give an intrinsic characterization of $C_{\sigma }$-uniqueness in terms of Skula-density and Scott closure, identifying precisely when a proper sub-dcpo can preserve the lattice of Scott-closed sets. Based on this characterization, we answer three open problems: (1) every power ${\text{J}}^I$ of Johnstone’s dcpo is $C_{\sigma }$-unique; (2) $C_{\sigma }$-uniqueness is not preserved by binary products, even when both factors and their product are sober; and (3) a sober countable complete lattice need not be $C_{\sigma }$-unique, even when it is a frame.