Direct product decompositions of lattices, closures and relation schemes
Leonid O. Libkin · Discrete Mathematics · 1993
In this paper we study the direct product decompositions of closure operations and lattices of closed sets. We characterize the direct product decompositions of lattices of closed sets in terms of closure operations, and find those decompositions of lattices which correspond to the decompositions of closures. If a closure on a finite set is represented by its implication base (i.e. a binary relation on the powerset), we construct a polynomial algorithm to find its direct product decompositions. The main characterization theorem is also applied to define direct product decompositions of relational database schemes and to find out what properties of relational databases and schemes are preserved under the decompositions.