Lattices of Irreducibly-derived Closed Sets

摘要

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 posetX, the setΓSI(X)of all irreducibly-derived Scott-closed sets (for short,SI-closed sets) ordered by inclusion forms a complete lattice. We introduce the notions ofCSI-continuous posets andCSI-prealgebraic posets and study their properties. We also introduce theSI-dominated posets and show that for any twoSI-dominated posetsXandY,X?Yif and only if theSI-closed set lattices above them are isomorphic. At last, we show that the category of strong complete posets withSI-continuous maps is Cartesian-closed.