Theory of granular lattices and its applications

Based on the concept of components of elements of a complete lattice this paper proposed the theory of granular lattices and proved that a topological space X is locally connected if and only if the lattice Ω(X) of its topology is a granular lattice. Moreover, this paper discussed compactness of abstract semantics on certain granular lattices and proved that the semantics is compact if it is continuous in a natural sense, and as a corollary, the compactness of Lukasiewiczʹs semantics was clarified.