Representability of binary relations through fuzzy numbers

We analyse the representability of different classes of binary relations on a set by means of suitable fuzzy numbers. In particular, we show that symmetric triangular fuzzy numbers can be considered as the best codomain to represent interval orders. We also pay attention to the representability of other classes of acyclic binary relations.

[1]  James F. Allen.  Maintaining knowledge about temporal intervals , 1983, CACM.

[2]  Peter C. Fishburn,et al.  Preference Structures and Their Numerical Representations , 1999, Theor. Comput. Sci..

[3]  Madan M. Gupta,et al.  Introduction to Fuzzy Arithmetic , 1991 .

[4]  José Carlos R. Alcantud,et al.  Characterization of the existence of semicontinuous weak utilities , 1999 .

[5]  Juan Carlos Candeal,et al.  Existence of homogeneous representations of interval orders on a cone in a topological vector space , 2005, Soc. Choice Welf..

[6]  Ramón Fuentes-González,et al.  The study of the interval-valued contexts , 2001, Fuzzy Sets Syst..

[7]  Juan Carlos Candeal,et al.  Numerical representability of semiorders , 2002, Math. Soc. Sci..

[8]  E. Induráin,et al.  Representability of Interval Orders , 1998 .

[9]  G. Cantor,et al.  Beiträge zur Begründung der transfiniten Mengenlehre , 1895 .

[10]  G. Mehta.  Preference and utility , 1998 .

[11]  A. Tversky.  Intransitivity of preferences. , 1969 .

[12]  G. Fechner.  Elemente der Psychophysik , 1998 .

[13]  Rudolf F. Albrecht,et al.  Topological interpretation of fuzzy sets and intervals , 2003, Fuzzy Sets Syst..

[14]  A. Kaufmann,et al.  Introduction to fuzzy arithmetic : theory and applications , 1986 .

[15]  Ron Shamir,et al.  Satisfiability Problems on Intervals and Unit Intervals , 1997, Theor. Comput. Sci..

[16]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[17]  P. Fishburn.  Interval representations for interval orders and semiorders , 1973 .

[18]  Alexis Tsoukiàs,et al.  A Characterization of PQI Interval Orders , 2003, Discret. Appl. Math..

[19]  Jaap Van Brakel,et al.  Foundations of measurement , 1983 .

[20]  Wayne Shafer,et al.  The Nontransitive Consumer , 1974 .

[21]  Gerhard Herden,et al.  The Debreu Gap Lemma and some generalizations , 2004 .

[22]  R. Luce.  Semiorders and a Theory of Utility Discrimination , 1956 .

[23]  Stefan Friedrich,et al.  Topology , 2019, Arch. Formal Proofs.

[24]  Jean-Paul Doignon,et al.  On realizable biorders and the biorder dimension of a relation , 1984 .

[25]  R. Agaev,et al.  Interval choice: classic and general cases , 1993 .

[26]  G. Cantor.  Beiträge zur Begründung der transfiniten Mengenlehre , 1897 .

[27]  Taner Bilgiç,et al.  Fusing Interval Preferences , 2001 .

[28]  V. Novák.  Fuzzy sets and their applications , 1989 .

[29]  Peter C. Fishburn,et al.  Utility theory for decision making , 1970 .

[30]  Peter C. Fishburn,et al.  LEXICOGRAPHIC ORDERS, UTILITIES AND DECISION RULES: A SURVEY , 1974 .

[31]  K. Arrow,et al.  General Competitive Analysis , 1971 .

[32]  P. Swistak.  Some representation problems for semiorders , 1980 .

[33]  D. Bridges,et al.  Representations of Preferences Orderings , 1995 .

[34]  D. Dubois,et al.  Operations on fuzzy numbers , 1978 .

[35]  G. Bosi,et al.  Representing preferences with nontransitive indifference by a single real-valued function☆ , 1995 .

[36]  K. Arrow,et al.  Social Choice and Individual Values , 1951 .

[37]  Johann Pfanzagl,et al.  Theory of measurement , 1970 .

[38]  Patrick Suppes,et al.  Foundations of measurement , 1971 .

[39]  李幼升,et al.  Ph , 1989 .

[40]  A. Beardon,et al.  The non-existence of a utility function and the structure of non-representable preference relations , 2002 .

[41]  Ramon E. Moore,et al.  Interval analysis and fuzzy set theory , 2003, Fuzzy Sets Syst..

[42]  F. Roberts.  Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences: Measurement Theory , 1984 .

[43]  Peter C. Fishburn,et al.  Intransitive Indifference in Preference Theory: A Survey , 1970, Oper. Res..

[44]  P. Hammond,et al.  Handbook of Utility Theory , 2004 .

[45]  Juan Carlos Candeal,et al.  Numerical Representations of Interval Orders , 2001, Order.

[46]  D. Bridges.  Numerical representation of intransitive preferences on a countable set , 1983 .

[47]  Taner Bilgiç,et al.  Interval-valued preference structures , 1998, Eur. J. Oper. Res..

[48]  P. Fishburn.  Intransitive indifference with unequal indifference intervals , 1970 .

[49]  Donald W. Katzner.  Static Demand Theory , 1970 .