A lemma on closures and its application to modularity in logic programming semantics

This note points out a lemma on closures of monotonic increasing functions and shows how it is applicable to decomposition and modularity for semantics defined as the least fixedpoint of some monotonic function. In particular it applies to numerous semantics of logic programs. An appendix addresses the fixedpoints of (possibly non-monotonic) functions that are sandwiched between functions with the same fixedpoints.