Fuzzy Mathematics

In the mid-1960's I had the pleasure of attending a talk by Lotfi Zadeh at which he presented some of his basic (and at the time, recent) work on fuzzy sets. Lotfi's algebra of fuzzy subsets of a set struck me as very nice; in fact, as a graduate student in the mid-1950's, I had suggested similar ideas about continuous-truth-valued propositional calculus (inffor "and", sup for "or") to my advisor, but he didn't go for it (and in fact, confused it with the foundations of probability theory), so I ended up writing a thesis in a more conventional area of mathematics (differential algebra). I especially enjoyed Lotfi's discussion of fuzzy convexity; I remember talking to him about possible ways of extending this work, but I didn't pursue this at the time. I have elsewhere told the story of how, when I saw C. L. Chang's 1968 paper on fuzzy topological spaces, I was impelled to try my hand at fuzzi­ fying algebra. This led to my 1971 paper "Fuzzy groups", which became the starting point of an entire literature on fuzzy algebraic structures. In 1974 King-Sun Fu invited me to speak at a U. S. -Japan seminar on Fuzzy Sets and their Applications, which was to be held that summer in Berkeley.

Contenu

1 Fuzzy Subsets.- 1.1 Fuzzy Relations.- 1.2 Operations on Fuzzy Relations.- 1.3 Reflexivity, Symmetry and Transitivity.- 1.4 Pattern Classification Based on Fuzzy Relations.- 1.5 Advanced Topics on Fuzzy Relations.- 1.6 References.- 2 Fuzzy Graphs.- 2.1 Paths and Connectedness.- 2.2 Clusters.- 2.3 Cluster Analysis and Modeling of Information Networks.- 2.4 Connectivity in Fuzzy Graphs.- 2.5 Application to Cluster Analysis.- 2.6 Operations on Fuzzy Graphs.- 2.7 Fuzzy Intersection Equations.- 2.8 Fuzzy Graphs in Database Theory.- 2.9 References.- 3 Fuzzy Topological Spaces.- 3.1 Topological Spaces.- 3.2 Metric Spaces and Normed Linear Spaces.- 3.3 Fuzzy Topological Spaces.- 3.4 Sequences of Fuzzy Subsets.- 3.5 F-Continuous Functions.- 3.6 Compact Fuzzy Spaces.- 3.7 Iterated Fuzzy Subset Systems.- 3.8 Chaotic Iterations of Fuzzy Subsets.- 3.9 Starshaped Fuzzy Subsets.- 3.10 References.- 4 Fuzzy Digital Topology.- 4.1 Introduction.- 4.2 Crisp Digital Topology.- 4.3 Fuzzy Connectedness.- 4.4 Fuzzy Components.- 4.5 Fuzzy Surroundedness.- 4.6 Components, Holes, and Surroundedness.- 4.7 Convexity.- 4.8 The Sup Projection.- 4.9 The Integral Projection.- 4.10 Fuzzy Digital Convexity.- 4.11 On Connectivity Properties of Grayscale Pictures.- 4.12 References.- 5 Fuzzy Geometry.- 5.1 Introduction.- 5.2 The Area and Perimeter of a Fuzzy Subset.- 5.3 The Height, Width and Diameter of a Fuzzy Subset.- 5.4 Distances Between Fuzzy Subsets.- 5.5 Fuzzy Rectangles.- 5.6 A Fuzzy Medial Axis Transformation Based on Fuzzy Disks.- 5.7 Fuzzy Triangles.- 5.8 Degree of Adjacency or Surroundedness.- 5.9 Image Enhancement and Thresholding Using Fuzzy Compactness.- 5.10 Fuzzy Plane Geometry: Points and Lines.- 5.11 Fuzzy Plane Geometry: Circles and Polygons.- 5.12 Fuzzy Plane Projective Geometry.- 5.13 A Modified Hausdorff Distance Between Fuzzy Subsets..- 5.14 References.- 6 Fuzzy Abstract Algebra.- 6.1 Crisp Algebraic Structures.- 6.2 Fuzzy Substructures of Algebraic Structures.- 6.3 Fuzzy Submonoids and Automata Theory.- 6.4 Fuzzy Subgroups, Pattern Recognition and Coding Theory.- 6.5 Free Fuzzy Monoids and Coding Theory.- 6.6 Formal Power Series, Regular Fuzzy Languages, and Fuzzy Automata.- 6.7 Nonlinear Systems of Equations of Fuzzy Singletons.- 6.8 Localized Fuzzy Subrings.- 6.9 Local Examination of Fuzzy Intersection Equations.- 6.10 More on Coding Theory.- 6.11 Other Applications.- 6.12 References.- List of Figures.- List of Tables.- List of Symbols.