## Fuzzy Discrete StructuresThis ambitious exposition by Malik and Mordeson on the fuzzification of discrete structures not only supplies a solid basic text on this key topic, but also serves as a viable tool for learning basic fuzzy set concepts "from the ground up" due to its unusual lucidity of exposition. While the entire presentation of this book is in a completely traditional setting, with all propositions and theorems provided totally rigorous proofs, the readability of the presentation is not compromised in any way; in fact, the many ex cellently chosen examples illustrate the often tricky concepts the authors address. The book's specific topics - including fuzzy versions of decision trees, networks, graphs, automata, etc. - are so well presented, that it is clear that even those researchers not primarily interested in these topics will, after a cursory reading, choose to return to a more in-depth viewing of its pages. Naturally, when I come across such a well-written book, I not only think of how much better I could have written my co-authored monographs, but naturally, how this work, as distant as it seems to be from my own area of interest, could nevertheless connect with such. Before presenting the briefest of some ideas in this direction, let me state that my interest in fuzzy set theory (FST) has been, since about 1975, in connecting aspects of FST directly with corresponding probability concepts. One chief vehicle in carrying this out involves the concept of random sets. |

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### Contents

Fuzzy Logic Functions | 1 |

12 Relations | 2 |

13 Functions | 5 |

14 Fuzzy Sets | 7 |

15 Semigroups | 8 |

16 Fuzzy Logic | 12 |

17 Fuzzy Functions and Decomposition | 16 |

18 Solution of Fuzzy Logic Inequalities | 33 |

44 References | 134 |

Fuzzy Machines Languages and Grammars | 139 |

52 Irreducibility and Minimality | 145 |

53 On Reductions of Maximin Machines | 148 |

54 ContextFree MaxProduct Grammars | 165 |

55 ContextFree Fuzzy Languages | 168 |

56 Deterministic Acceptors of Regular Fuzzy Languages | 173 |

57 Fuzzy Languages on a Free Monoid | 179 |

19 References | 40 |

Decision Trees | 41 |

22 Fuzzy Decision Tree Algorithms | 44 |

23 Analysis of the BBB Algorithm | 53 |

24 References | 55 |

Networks | 57 |

32 A Maximum Flow Algorithm | 59 |

33 The Max Flow Min Cut Theorem | 65 |

34 Maximum Flow in a Network with Fuzzy Arc Capacities | 67 |

35 The Maximum Flow with Integer Values | 74 |

36 Integer Flows in Network with TwoSided Fuzzy Capacity Constraints | 76 |

37 RealValued Flows in a Network with Fuzzy Arc Capacities | 83 |

38 Petri Nets | 91 |

39 Fuzzy Petri Nets for RuleBased Decisionmaking | 95 |

310 References | 103 |

Fuzzy Graphs and Shortest Paths | 107 |

42 Analysis of the Fuzzy Path Models | 116 |

43 On Valuation and Optimization Problems | 120 |

58 Algebraic Character and Properties of FRegular Languages | 182 |

59 References | 195 |

Algebraic Fuzzy Automata | 197 |

62 Homomorphisms | 201 |

63 Admissible Relations | 203 |

64 Fuzzy Transformation Semigroups | 205 |

65 Submachines | 209 |

66 Retrievability Separability and Connectivity | 213 |

67 Decomposition of Fuzzy Finite State Machines | 216 |

68 Admissible Partitions | 219 |

69 On Fuzzy Recognizers | 227 |

610 Minimal Fuzzy Recognizers | 239 |

611 References | 245 |

247 | |

255 | |

261 | |

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### Common terms and phrases

admissible partition algebra algorithm arc capacities automata called capacity constraints congruence relation Consider convex maximin decision tree decision value defined Definition denote deterministic disjunctive normal form edge equivalence relation Example exists F-adjunctive language F-dense ffsm Figure finite state machine flow xv follows from Proposition free monoid free semigroup fuzzy finite fuzzy function fuzzy graph fuzzy language fuzzy logic fuzzy numbers fuzzy recognizer Fuzzy Sets fuzzy shortest path fuzzy subset fuzzy truth G Qd G Qi Go to Step Greibach normal form Hence homomorphism implies integer irreducible ISBN Lemma len(x len(y Let f matrix max-product maximal maximum flow MPSM MSLM neuron node normal form optimal flow partition of Q Petri nets primary submachine proof follows real numbers semigroup set of vertices Sets and Systems shortest path problem solution statewise minimal strongly connected submodular Suppose Theorem truth token vertex