Stone duality and representation of stable domain

In this paper, the author studies the Stone duality and representation of L-domain w.r.t. stable functions. Two basic notions are introduced, one is D-semilattice and the other is semitopological system. The author first gives the representation of stable D-lattice, then establishes the representation theorem of L-domain and gives the Stone duality of L-domains w.r.t. stable functions in the scheme of lattice. The author introduces the theory of semitopological systems, which is a generalization of Vickersʹ topological systems. By using it, the author gives the Stone duality of L-domains w.r.t. stable functions and that of Scott domains w.r.t. functions preserving the directed sups and meets of nonempty subsets in the scheme of topology.