A Stone-type duality for sT0sT0 stratified Alexandrov L-topological spaces

We show that for every complete lattice A  , both the set of completely prime elements and the set of completely coprime elements are one-to-one corresponding to CH(A)CH(A), the set of complete lattice homomorphisms from A to the two-element lattice 2. A complete lattice is called completely generated, a cg-lattice for short, if it is generated by the set of completely prime elements, or equivalently, by the set of completely coprime elements. Then we restudy the duality between the category of T0T0 Alexandrov topological spaces and the category of cg-lattices by means of CH(A)CH(A). With these preparations, for a frame L   as the truth value table, we introduce sT0sT0 separation axiom for stratified Alexandrov L  -topological spaces, and finally establish a duality between the category of sT0sT0 stratified Alexandrov L-topological spaces and the category of completely generated complete L  -ordered sets. We also investigate some properties of the sT0sT0 axiom, for example, it is hereditary by closed subspaces and productive.