Adaricheva, KiraNation, J.B.2016-02-092016-02-092013Adaricheva Kira, Nation J.B.; 2013; On implicational bases of closure system with unique critical sets; arXiv.orghttp://nur.nu.edu.kz/handle/123456789/1209We show that every optimum basis of a nite closure system, in D. Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce the K-basis of a general closure system, which is a re nement of the canonical basis of V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain it. We study closure systems with unique critical sets, and some subclasses of these where the K-basis is unique. A further re nement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Thus, closure systems without D-cycles can be e ectively recognized. While the E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete problemenAttribution-NonCommercial-ShareAlike 3.0 United StatesResearch Subject Categories::MATHEMATICSfinite closure systemArticle