# zbMATH — the first resource for mathematics

On the cycle-transitivity of the mutual rank probability relation of a poset. (English) Zbl 1256.06001
Summary: 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.

##### MSC:
 06A06 Partial orders, general 06A05 Total orders
Full Text:
##### References:
 [1] Brüggemann, R.; Restrepo, G.; Voigt, K., Structure-fate relationships of organic chemicals derived from the software packages E4CHEM and WHASSE, Journal of chemical information modelling, 46, 894-902, (2006) [2] Brüggemann, R.; Simon, U.; Mey, S., Estimation of averaged ranks by extended local partial order models, MATCH communications in mathematical and in computer chemistry, 54, 489-518, (2005) · Zbl 1082.92503 [3] Brüggemann, R.; Sørensen, P.; Lerche, D.; Carlsen, L., Estimation of averaged ranks by a local partial order model, Journal of chemical information and computer science, 4, 618-625, (2004) [4] Brüggemann, R.; Voigt, K., Basic principles of Hasse diagram technique in chemistry, Combinatorial chemistry & high throughput screening, 11, 756-769, (2008) [5] Chiclana, P.; Herrera-Viedma, E.; Alonso, S.; Herrera, F., Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity, IEEE transactions on fuzzy systems, 17, 14-23, (2009) [6] De Baets, B.; De Meyer, H., Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity, Fuzzy sets and systems, 152, 249-270, (2005) · Zbl 1114.91031 [7] De Baets, B.; De Meyer, H., On the cycle-transitive comparison of artificially coupled random variables, International journal of approximate reasoning, 47, 306-322, (2008) · Zbl 1201.60014 [8] De Baets, B.; De Meyer, H.; De Schuymer, B.; Jenei, S., Cyclic evaluation of transitivity of reciprocal relations, Social choice and welfare, 26, 217-238, (2006) · Zbl 1158.91338 [9] B. De Baets, B. De Schuymer, H. De Meyer, A graph-theoretical characterization of cycle-transitivity w.r.t. commutative dual quasi-copulas, in preparation. · Zbl 1115.60019 [10] De Loof, K.; De Baets, B.; De Meyer, H., Counting linear extension majority cycles in posets on up to 13 points, Computers and mathematics with applications, 59, 1541-1547, (2010) · Zbl 1189.06001 [11] K. De Loof, B. De Baets, H. De Meyer, Cycle-free cuts of mutual rank probability relations, submitted for publication. · Zbl 1323.06001 [12] De Loof, K.; De Baets, B.; De Meyer, H.; Brüggemann, R., A Hitchhiker’s guide to poset ranking, Combinatorial chemistry and high throughput screening, 11, 734-744, (2008) [13] De Loof, K.; De Meyer, H.; De Baets, B., Exploiting the lattice of ideals representation of a poset, Fundamenta informaticae, 71, 309-321, (2006) · Zbl 1110.06001 [14] De Meyer, H.; De Baets, B.; De Schuymer, B., On the transitivity of comonotonic and countermonotonic comparison of random variables, Journal of multivariate analysis, 98, 177-193, (2007) · Zbl 1114.60018 [15] De Schuymer, B.; De Meyer, H.; De Baets, B., Cycle-transitive comparison of independent random variables, Journal of multivariate analysis, 96, 352-373, (2005) · Zbl 1087.60018 [16] De Schuymer, B.; De Meyer, H.; De Baets, B., Optimal strategies for equal-sum dice games, Discrete applied mathematics, 154, 2565-2576, (2006) · Zbl 1255.91009 [17] De Schuymer, B.; De Meyer, H.; De Baets, B., Extreme copulas and the comparison of ordered lists, Theory and decision, 62, 195-217, (2007) · Zbl 1115.60019 [18] De Schuymer, B.; De Meyer, H.; De Baets, B.; Jenei, S., On the cycle-transitivity of the dice model, Theory and decision, 54, 261-285, (2003) · Zbl 1075.60011 [19] Díaz, S.; Montes, S.; De Baets, B., Transitivity bounds in additive fuzzy preference structures, IEEE transactions on fuzzy systems, 15, 275-286, (2007) [20] Fishburn, P., Binary choice probabilities: varieties of stochastic transitivity, Journal of mathematical psychology, 10, 327-352, (1973) · Zbl 0277.92008 [21] Fishburn, P., On the family of linear extensions of a partial order, Journal of combinatorial theory series B, 17, 240-243, (1974) · Zbl 0274.06003 [22] Fishburn, P., Linear extension majority graphs of partial orders, Journal of combinatorial theory series B, 21, 65-70, (1976) · Zbl 0294.06001 [23] Fishburn, P., Proportional transitivity in linear extensions of ordered sets, Journal of combinatorial theory series B, 41, 48-60, (1986) · Zbl 0566.06002 [24] Herrera-Viedma, E.; Herrera, F.; Chiclana, F.; Luque, M., Some issues on consistency of fuzzy preference relations, European journal of operational research, 154, 98-109, (2004) · Zbl 1099.91508 [25] Kahn, J.; Yu, Y., Log-concave functions and poset probabilities, Combinatorica, 18, 85-99, (1998) · Zbl 0928.52006 [26] Kislitsyn, S., Finite partially ordered sets and their associated sets of permutations, Matematicheskiye zametki, 4, 511-518, (1968) [27] Klement, E.; Mesiar, R.; Pap, E., Triangular norms, trends in logic, studia logica library, vol. 8, (2000), Kluwer Academic Publishers Dordrecht [28] Lerche, D.; Sørensen, P.; Brüggemann, R., Improved estimation of the ranking probabilities in partial orders using random linear extensions by approximation of the mutual ranking probability, Journal of chemical information and computer science, 43, 1471-1480, (2003) [29] R. Nelsen, An introduction to copulas, Lecture Notes in Statistics, Vol. 139, second ed., Springer-Verlag, New York, 2005. · Zbl 1079.60021 [30] Switalski, Z., General transitivity conditions for fuzzy reciprocal preference matrices, Fuzzy sets and systems, 137, 85-100, (2003) · Zbl 1052.91033 [31] Tanino, T., Fuzzy preference relations in group decision making, (), 54-71 [32] W. Waegeman, B. De Baets, A transitivity analysis of bipartite rankings in pairwise multi-class classification, Information Sciences, submitted for publication. · Zbl 1204.62103 [33] Yu, Y., On proportional transitivity of ordered sets, Order, 15, 87-95, (1998) · Zbl 0912.06005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.