Lattices of Scott-closed sets. (English) Zbl 1212.06010

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.


06B35 Continuous lattices and posets, applications
06A06 Partial orders, general
06A12 Semilattices
06B23 Complete lattices, completions
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