On the cycle-transitivity of the mutual rank probability relation of a poset

The mutual rank probability relation associated with a finite poset is a reciprocal relation expressing the probability that a given element succeeds another one in a random linear extension of that poset. We contribute to the characterization of the transitivity of this mutual rank probability relation, also known as proportional probabilistic transitivity, by situating it between strong stochastic transitivity and moderate product transitivity. The methodology used draws upon the cycle-transitivity framework, which is tailor-made for describing the transitivity of reciprocal relations.

[1]  Rainer Brüggemann,et al.  Structure-Fate Relationships of Organic Chemicals Derived from the Software Packages E4CHEM and WHASSE , 2006, J. Chem. Inf. Model..

[2]  B. De Baets,et al.  On the transitivity of the comonotonic and countermonotonic comparison of random variables , 2007 .

[3]  Peter C. Fishburn.  On the family of linear extensions of a partial order , 1974 .

[4]  R. Nelsen.  An Introduction to Copulas , 1998 .

[5]  Tetsuzo Tanino,et al.  Fuzzy Preference Relations in Group Decision Making , 1988 .

[6]  Rainer Brüggemann,et al.  Improved Estimation of the Ranking Probabilities in Partial Orders Using Random Linear Extensions by Approximation of the Mutual Ranking Probability , 2003, J. Chem. Inf. Comput. Sci..

[7]  Rainer Brüggemann,et al.  Basic principles of Hasse diagram technique in chemistry. , 2008, Combinatorial chemistry & high throughput screening.

[8]  R. Nelsen.  An Introduction to Copulas (Springer Series in Statistics) , 2006 .

[9]  Zbigniew Switalski,et al.  General transitivity conditions for fuzzy reciprocal preference matrices , 2003, Fuzzy Sets Syst..

[10]  Rainer Brüggemann,et al.  A hitchhiker's guide to poset ranking. , 2008, Combinatorial chemistry & high throughput screening.

[11]  Bernard De Baets,et al.  On the cycle-transitive comparison of artificially coupled random variables , 2008, Int. J. Approx. Reason..

[12]  Francisco Herrera,et al.  Some issues on consistency of fuzzy preference relations , 2004, Eur. J. Oper. Res..

[13]  Francisco Herrera,et al.  Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity , 2009, IEEE Transactions on Fuzzy Systems.

[14]  Bernard De Baets,et al.  A transitivity analysis of bipartite rankings in pairwise multi-class classification , 2010, Inf. Sci..

[15]  Bernard De Baets,et al.  Cyclic Evaluation of Transitivity of Reciprocal Relations , 2006, Soc. Choice Welf..

[16]  Yang Yu.  On Proportional Transitivity of Ordered Sets , 1998 .

[17]  Bernard De Baets,et al.  Exploiting the Lattice of Ideals Representation of a Poset , 2006, Fundam. Informaticae.

[18]  Bernard De Baets,et al.  Transitivity Bounds in Additive Fuzzy Preference Structures , 2007, IEEE Transactions on Fuzzy Systems.

[19]  B. De Baets,et al.  On the Cycle-Transitivity of the Dice Model , 2003 .

[20]  Radko Mesiar,et al.  Triangular Norms , 2000, Trends in Logic.

[21]  P. Fishburn.  Binary choice probabilities: on the varieties of stochastic transitivity , 1973 .

[22]  Jeff Kahn,et al.  Log-Concave Functions And Poset Probabilities , 1998, Comb..

[23]  Peter C. Fishburn.  Proportional transitivity in linear extensions of ordered sets , 1986, J. Comb. Theory, Ser. B.

[24]  B. De Baets,et al.  Cycle-transitive comparison of independent random variables , 2005 .

[25]  B. D. Baetsb,et al.  Optimal strategies for equal-sum dice games , 2006 .

[26]  B. De Baets,et al.  Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity , 2005, Fuzzy Sets Syst..

[27]  PETER C. FISHBURN.  On linear extension majority graphs of partial orders , 1976, J. Comb. Theory, Ser. B.

[28]  Rainer Brüggemann,et al.  Estimation of averaged ranks by extended local partial order models , .

[29]  Rainer Brüggemann,et al.  Estimation of Averaged Ranks by a Local Partial Order Model , 2004, J. Chem. Inf. Model..

[30]  B. Baets,et al.  Extreme Copulas and the Comparison of Ordered Lists , 2007 .

[31]  Bernard De Baets,et al.  Counting linear extension majority cycles in partially ordered sets on up to 13 elements , 2010, Comput. Math. Appl..