Ho, Weng Kin; Zhao, Dongsheng Lattices of Scott-closed sets. (English) Zbl 1212.06010 Commentat. Math. Univ. Carol. 50, No. 2, 297-314 (2009). Lattices of Scott-open sets and Scott-closed sets have been intensively studied since 1980. A basic result is the Characterization Theorem for domains, which states that a lattice \(L\) is a domain if and only if the lattice of Scott-open sets of \(L\) is completely distributive. A fundamental result, due to S. Papert, is that a complete lattice \(L\) is isomorphic to the lattice of closed subsets of a topological space \(X\) iff the co-primes of \(L\) are join-dense in \(L\).The authors extend the study of Scott-closed subsets to posets. A poset \(P\) is called a depo if every directed subposet of \(P\) has a supremum in \(P\). Let \(C(P)\) denote the lattice of Scott-closed subsets of a depo \(P\). It is known that \(C(P)\) is completely distributive iff \(P\) is continuous.Main results: (i) every \(C(P)\) is C-continuous; (ii) a complete lattice \(L\) is isomorphic to \(C(P)\) for a complete semilattice \(P\) iff \(L\) is weak-stable and C-algebraic; (iii) for any complete semilattices \(P\) and \(Q\), \(P\) and \(Q\) are isomorphic iff \(C(P)\) and \(C(Q)\) are isomorphic. Finally, the functor from \(P\) to \(C(P)\) is extended to a left adjoint functor from the category of depos to the category of C-prealgebraic lattices. Reviewer: Ivan Chajda (Olomouc) Cited in 1 ReviewCited in 16 Documents MSC: 06B35 Continuous lattices and posets, applications 06A06 Partial orders, general 06A12 Semilattices 06B23 Complete lattices, completions Keywords:domain; complete semilattice; Scott-closed set; C-algebraic lattice PDFBibTeX XMLCite \textit{W. K. Ho} and \textit{D. Zhao}, Commentat. Math. Univ. Carol. 50, No. 2, 297--314 (2009; Zbl 1212.06010) Full Text: EuDML EMIS