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It is well known that the linear extension majority (LEM) relation of a poset of size $n\ge 9$ 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...