1: Binary Relations: Definitions, Representations, Basic Properties -- 1.1. Binary relations -- 1.2. Graph representation of binary relations -- 1.3. Coding the binary relations -- 1.4. Matrix representation of binary relations -- 1.5. Basic properties of binary relations -- 1.6. Particular binary relations -- 1.7. Graph interpretation of the properties -- 1.8. Algebraic interpretation of the properties -- 1.9. References -- 2: The Concept of Preference Structure -- 2.1. Preference, indifference, incomparability -- 2.2. Preference structure -- 2.3. Important agreement -- 2.4. Characteristic relation of a preference structure -- 2.5. Graph representation of a preference structure -- 2.6. Coding the preference structure -- 2.7. Example -- 2.8. References -- 3: Usual Preference Structures -- 3.1. Tournament structure -- 3.2. Total order structure -- 3.3. Weak order structure -- 3.4. Total interval order structure -- 3.5. Total semiorder structure -- 3.6. Partial order structure -- 3.7. Quasi order structure -- 3.8. References -- 4: Two New Preference Structures -- 4.1. Partial interval order structure -- 4.2. Partial semiorder structure -- 4.3. References -- 5: Complete Valued Preference Structures -- 5.1. Definition -- 5.2. Important remark -- 5.3. Particular case -- 5.4. Graph representation -- 5.5. Matridal representation -- 5.6. Particular complete valued preference structures -- 5.7. Binary relations and various properties related to a complete valued preference structure -- 5.8. Characterizations of the families defined in section 5.6.. -- 5.9. Functional representation of a valued preference structure -- 5.10. Roberts homogeneous families of semiorders -- 5.11. Families of weak orders -- 5.12. Summary -- 5.13. Examples -- 5.14. References -- 6: Complete Two-Valued Preference Structures -- 6.1. Introduction -- 6.2. Two-valued preference structures with constant thresholds -- 6.3. Example -- 6.4. References.

The following scheme summarizes the different families introduced in this chapter and the connections between them. Family of interval orders f Row-homogeneous Column-homogeneous Family of family of interval semi orders family of interval orders orders Homogeneous family of i nterva 1 orders Homogeneous family of semi orders Family of weak orders 85 5.13. EXAMPLES We let to the reader the verification of the following assertions. Example 1 is a family of interval orders which is neither row-homogeneous nor column-homogeneous. Example 2 is a column-homogeneous family of interval orders which is not row-homogeneous but where each interval order is a semiorder. Example 3 is an homogeneous family of interval orders which are not semiorders. Example 4 is an homogeneous family of semi orders . . 8 ~ __ --,b ~---i>---_ C a .2 d c Example Example 2 .8 .6 c .5 a 0 a d Example 3 Example 4 5.14. REFERENCES DOIGNON. J.-P •• Generalizations of interval orders. in E. Degreef and J. Van Buggenhaut (eds). T~ndS in MathematiaaZ PsyahoZogy. Elsevier Science Publishers B.V. (North-Holland), Amsterdam, 1984. FISHBURN. P.C., Intransitive indifference with unequal indifference intervals. J. Math. Psyaho.~ 7 (1970) 144-149. FISHBURN. P.C., Binary choice probabilities: on the varieties of stochastic transitivity. J. Math. Psyaho.~ 10 (1973) 327-352.