Cycle-free cuts of mutual rank probability relations

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 αm such that the crisp relation obtained from the mutual rank probability relation by setting to 0 its elements smaller than or equal to αm, and to 1 its other elements, is free from cycles of length m. In a first part, theoretical upper bounds for αm 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 α4 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.