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Cycle-free cuts of mutual rank probability relations. (English) Zbl 1323.06001
Summary: It is well known that the linear extension majority (LEM) relation of a poset of size $$n\geq 9$$ can contain cycles. In this paper we are interested in obtaining minimum cutting levels $$\alpha_m$$ such that the crisp relation obtained from the mutual rank probability relation by setting to 0 its elements smaller than or equal to $$\alpha_m$$, and to 1 its other elements, is free from cycles of length $$m$$. In a first part, theoretical upper bounds for $$\alpha_m$$ are derived using known transitivity properties of the mutual rank probability relation. Next, we experimentally obtain minimum cutting levels for posets of size $$n\leq 13$$. 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.

##### MSC:
 06A06 Partial orders, general 06A07 Combinatorics of partially ordered sets
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##### References:
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