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Cycle-free cuts of mutual rank probability relations

Karel De Loof (UGent) , Bernard De Baets (UGent) and Hans De Meyer (UGent)
(2014) KYBERNETIKA. 50(5). p.814-837
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Abstract
It is well known that the linear extension majority (LEM) relation of a poset of size n >= 9 can contain cycles. In this paper we are interested in obtaining minimum cutting levels am such that the crisp relation obtained from the mutual rank probability relation by setting to 0 its elements smaller than or equal to am, and to 1 its other elements, is free from cycles of length m. In a first part, theoretical upper bounds for am are derived using known transitivity properties of the mutual rank probability relation. Next, we experimentally obtain minimum cutting levels for posets of size n <= 13. We study the posets requiring these cutting levels in order to have a cycle-free strict cut of their mutual rank probability relation. Finally, a lower bound for the minimum cutting level alpha(4) is computed. To accomplish this, a family of posets is used that is inspired by the experimentally obtained 12-element poset requiring the highest cutting level to avoid cycles of length 4.
Keywords
cycle-free cut, minimum cutting level, EXTENSION MAJORITY CYCLES, PARTIAL ORDERS, LINEAR EXTENSIONS, PROPORTIONAL TRANSITIVITY, SETS, POSET, linear extension majority cycle, mutual rank probability relation, partially ordered set

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MLA
De Loof, Karel, et al. “Cycle-Free Cuts of Mutual Rank Probability Relations.” KYBERNETIKA, vol. 50, no. 5, 2014, pp. 814–37, doi:10.14736/kyb-2014-5-0814.
APA
De Loof, K., De Baets, B., & De Meyer, H. (2014). Cycle-free cuts of mutual rank probability relations. KYBERNETIKA, 50(5), 814–837. https://doi.org/10.14736/kyb-2014-5-0814
Chicago author-date
De Loof, Karel, Bernard De Baets, and Hans De Meyer. 2014. “Cycle-Free Cuts of Mutual Rank Probability Relations.” KYBERNETIKA 50 (5): 814–37. https://doi.org/10.14736/kyb-2014-5-0814.
Chicago author-date (all authors)
De Loof, Karel, Bernard De Baets, and Hans De Meyer. 2014. “Cycle-Free Cuts of Mutual Rank Probability Relations.” KYBERNETIKA 50 (5): 814–837. doi:10.14736/kyb-2014-5-0814.
Vancouver
1.
De Loof K, De Baets B, De Meyer H. Cycle-free cuts of mutual rank probability relations. KYBERNETIKA. 2014;50(5):814–37.
IEEE
[1]
K. De Loof, B. De Baets, and H. De Meyer, “Cycle-free cuts of mutual rank probability relations,” KYBERNETIKA, vol. 50, no. 5, pp. 814–837, 2014.
@article{6846755,
  abstract     = {{It is well known that the linear extension majority (LEM) relation of a poset of size n >= 9 can contain cycles. In this paper we are interested in obtaining minimum cutting levels am such that the crisp relation obtained from the mutual rank probability relation by setting to 0 its elements smaller than or equal to am, and to 1 its other elements, is free from cycles of length m. In a first part, theoretical upper bounds for am are derived using known transitivity properties of the mutual rank probability relation. Next, we experimentally obtain minimum cutting levels for posets of size n <= 13. We study the posets requiring these cutting levels in order to have a cycle-free strict cut of their mutual rank probability relation. Finally, a lower bound for the minimum cutting level alpha(4) is computed. To accomplish this, a family of posets is used that is inspired by the experimentally obtained 12-element poset requiring the highest cutting level to avoid cycles of length 4.}},
  author       = {{De Loof, Karel and De Baets, Bernard and De Meyer, Hans}},
  issn         = {{0023-5954}},
  journal      = {{KYBERNETIKA}},
  keywords     = {{cycle-free cut,minimum cutting level,EXTENSION MAJORITY CYCLES,PARTIAL ORDERS,LINEAR EXTENSIONS,PROPORTIONAL TRANSITIVITY,SETS,POSET,linear extension majority cycle,mutual rank probability relation,partially ordered set}},
  language     = {{eng}},
  number       = {{5}},
  pages        = {{814--837}},
  title        = {{Cycle-free cuts of mutual rank probability relations}},
  url          = {{http://doi.org/10.14736/kyb-2014-5-0814}},
  volume       = {{50}},
  year         = {{2014}},
}
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