Formal concept analysis and lattice-valued Chu systems

  • Authors:

  • Jeffrey T. Denniston;Austin Melton;Stephen E. Rodabaugh

  • Affiliations:

  • Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA;Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA and Department of Computer Science, Kent State University, Kent, OH 44242, USA;College of Science, Technology, Engineering, Mathematics (STEM), Youngstown State University, Youngstown, OH 44555-3347, USA

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2013

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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.