A theory of finite closure spaces based on implications
The theory of functional dependencies in relational database schemes is a neat example of a miniaturized logic: Notions like syntax, semantics, inference are present and the Amstron axioms are a simple tool to explain such fundamental concepts as soundness and completeness. It comes therefore as a surprise to observe that the very same notion can be the starting point for a research program on the interface between combinatorics (matroid theory), group theory, geometry (lattices) and topology (closure operators). The author seems to be involved in a general abstract characterization of minimal (optimal) systems of functional dependencies. In relational database theory these minimal covers are well understood; finding them is hard (NP-complete problems in the neighborhood). However, since the resulting characterizations are not translated backwards into the language of the application areas the question remains whether the characterizations have any merit other than being mathematically proven results. The reviewer failed to understood the relevance of the various notions of closure involved in this study for the database applications. I abstain from commenting on their relevance for combinatorics and/or topology. I do consider it unlikely however that the author has succeeded in his quest to rebut the prevalent opinion that ``no interesting space theory can be developed on the general concept of a closure space''.
scientific article; zbMATH DE number 706489