Formal concept analysis and lattice-valued Chu systems

Abstract

This paper links formal concept analysis (FCA) both to order-theoretic developments in the theory of Galois connections and to Chu spaces or systems viewed as a common rubric for both topological systems and systems arising from predicate transformers in programming semantics [13]. These links are constructed for each of traditional FCA and $L$-FCA, where $L$ is a commutative residuated semiquantale. Surprising and important consequences include relationships between formal ($L$-)contexts and ($L$-)topological systems within the category of ($L$-)Chu systems, relationships justifying the categorical study of formal ($L$-)contexts and linking such study to ($L$-)Chu systems. Applications and potential applications are primary motivations, including several example classes of formal ($L$-)contexts induced from data mining notions. Throughout, categorical frameworks are given for FCA and lattice-valued FCA in which morphisms preserve the Birkhoff operators on which all the structures of FCA and lattice-valued FCA rest; and, further, the results of this paper show that, under very general conditions, these categorical frameworks are both sufficient and necessary for the “interchange” or “preservation” of ($L$-)concepts and ($L$-)protoconcepts, structures central to FCA and lattice-valued FCA.