## Semiorders: Properties, Representations, ApplicationsSemiorder is probably one of the most frequently ordered structures in science. It naturally appears in fields like psychometrics, economics, decision sciences, linguistics and archaeology. It explicitly takes into account the inevitable imprecisions of scientific instruments by allowing the replacement of precise numbers by intervals. The purpose of this book is to dissect this structure and to study its fundamental properties. The main subjects treated are the numerical representations of semiorders, the generalizations of the concept to valued relations, the aggregation of semiorders and their basic role in a general theoretical framework for multicriteria decision-aid methods. Audience: This volume is intended for students and researchers in the fields of decision analysis, management science, operations research, discrete mathematics, classification, social choice theory, and order theory, as well as for practitioners in the design of decision tools. |

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### Contents

FIRST PRESENTATION OF THE BASIC CONCEPTS | 7 |

12 Introduction to the concept of semiorder | 9 |

122 Strict complete order associated to a semiorder | 12 |

124 Example | 13 |

13 Introduction to the concept of interval order | 14 |

132 Example | 15 |

14 Interval graph and indifference graph | 16 |

15 Valued semiorder | 17 |

412 Semiorders as interval orders and the representation of interval orders | 86 |

413 Ordered structures associated with a steptype matrix | 88 |

414 Representations of all interval orders associated with a steptype matrix | 89 |

415 Minimal representation of an interval order | 91 |

416 Proofs of the theorems | 93 |

VALUED SEMIORDERS | 103 |

53 Graph and matrix representations of a valued relation | 104 |

54 Semiordered valued relation | 105 |

HISTORICAL REVIEW AND APPLICATIONS | 19 |

22 Semiorders in genetics | 22 |

23 Semiorders bandwidth and other difficult combinatorial problems | 25 |

24 Semiorders in information storage | 26 |

25 Semiorders in decisionaid | 27 |

26 Semiorders in scheduling | 28 |

28 Interval orders and semiorders in archaeology | 29 |

29 Interval orders and semiorders in psychology | 31 |

211 Semiorders in classification | 32 |

212 Semiorders and rough sets | 42 |

213 Semiorders and fuzzy sets | 43 |

214 Semiorders and a theory of evolution of rationality | 44 |

215 Appendix | 45 |

BASIC CONCEPTS AND DEFINITIONS | 49 |

32 Asymmetric and symmetric parts of a relation | 50 |

35 Complement converse and dual of a relation | 51 |

37 Semiorder | 52 |

39 Dual of a semiorder strict semiorder | 53 |

310 Complete preorder | 54 |

313 Complete order | 55 |

316 Complete preorders and strict complete order associated to a semiorder | 56 |

317 Graph and matrix representations of a semi order | 57 |

318 Example of semiorder | 58 |

319 Interval order | 60 |

321 Dual of an interval order strict interval order | 61 |

323 Graph and matrix representations of an interval order | 62 |

324 Example of interval order | 63 |

325 Proofs of the theorems | 65 |

MINIMAL REPRESENTATIONS | 71 |

42 Alternative definition of a finite semiorder | 72 |

44 An algorithm for finding the numerical representation of a semiorder | 73 |

46 Minimal representation of a semiorder | 74 |

47 Integrality of the minimal representation | 75 |

48 Maximal contrast property | 76 |

49 An example and the synthetic graph of a se miorder | 77 |

410 Noses and Hollows | 79 |

411 More about paths of SSG | 82 |

56 Example of a semiordered valued relation | 106 |

57 Minimal representation of a semiordered valued relation | 108 |

58 Ipsodual semiordered valued relation | 109 |

510 Example of an ipsodual semiordered valued relation | 111 |

513 Numerical representation of a linear semiordered valued relation | 112 |

514 Example of a linear semiordered valued relation | 113 |

517 Numerical representation of an ipsodual linear semiordered valued relation | 114 |

519 Minimal representation of an ipsodual semi ordered valued relation | 115 |

522 Minimal representation | 118 |

523 Proofs of the theorems | 119 |

AGGREGATION OF SEMIORDERS | 125 |

61 Arrows theorem for semiorders | 126 |

62 Lexicographic aggregation of semiorders | 127 |

622 First variant | 128 |

623 Second variant | 129 |

624 Third variant | 130 |

625 A note on strict preference | 131 |

626 A note on indifference | 132 |

627 Fourth variant | 133 |

628 Fifth variant | 135 |

63 Aggregation of semiorders by Bordas method | 136 |

632 Use of the concept of rank of an element in a graph without circuit | 137 |

633 Using scores | 139 |

64 The dominant aggregation paradigm applied to semiorders | 141 |

65 Theoretical results related to the overall evaluation approach | 144 |

66 The pairwise comparisons paradigm | 148 |

67 A general framework | 151 |

68 Aggregation of valued serniorders and families of semiorders | 157 |

69 Proofs of the theorems Proof of theorem | 160 |

MISCELLANEOUS | 167 |

72 Semiordered mixture sets | 168 |

74 Double threshold models | 169 |

76 Enumerating semiorders | 170 |

78 Indifference graphs and families of indifference graphs | 171 |

CONCLUSION | 173 |

### Other editions - View all

Semiorders: Properties, Representations, Applications Marc Pirlot,P. Vincke No preview available - 2013 |

Semiorders: Properties, Representations, Applications Marc Pirlot,P. Vincke No preview available - 2010 |

### Common terms and phrases

aggregation of semiorders alternatives arcs aSdb asymmetric asymmetric relation binary relation Bouyssou chain of semiorders chapter characterization circuit complete preorder consider constant thresholds corresponding decision decision-aid defined denote diagonal Doignon e-representation equal equivalence relation example exists family of semiorders figure finite set function g gene given in section hence hp(a hp(b implies indifference graph integer interval graph irreflexive ISBN linear semiordered valued lines and columns mathematical minimal representation nodes nonnegative constant numerical representation obtained overall preference P-arcs pair Pirlot possible potential function preference relation problem procedure proof is given Proof of theorem rank real-valued function reflexive and complete reflexive relation represented resp rough sets satisfies semiordered valued relation step-type matrix strict complete order strict preference strict semiorder strongly connected component structure theory threshold q ultrametric valued functions valued graph valued semiorder Vincke weak order

### Popular passages

Page ii - In particular, formal treatment of social phenomena, the analysis of decision making, information theory and problems o ("inference will be central themes of this part of the library. Besides theoretical results, empirical investigations and the testing of theoretical models of real world problems will be subjects of interest. In addition to emphasizing interdisciplinary communication, the series will seek to support the rapid dissemination of recent results.

Page 178 - Some applications of graph theory and related nonmetric techniques to problems of approximate seriation: The case of symmetric proximity measures. The British Journal of Mathematical and Statistical Psychology.