Abstract
This paper pursues an investigation on the lattices of irreducibly-derived closed sets initiated by Zhao and Ho (2015). This time we focus the closed set lattice arising from the irreducibly-derived topology of Scott topology. For a poset X, the set ΓSI(X) of all irreducibly-derived Scott-closed sets (for short, SI-closed sets) ordered by inclusion forms a complete lattice. We introduce the notions of CSI-continuous posets and CSI-prealgebraic posets and study their properties. We also introduce the SI-dominated posets and show that for any two SI-dominated posets X and Y, X≅Y if and only if the SI-closed set lattices above them are isomorphic. At last, we show that the category of strong complete posets with SI-continuous maps is Cartesian-closed.
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