Lattices of Irreducibly-derived Closed Sets

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 Γ S I ( 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 C S I -continuous posets and C S I -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.