Formal concept analysis and lattice-valued Chu systems

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.