2 research outputs found
Cycle-free cuts of mutual rank probability relations
summary:It is well known that the linear extension majority (LEM) relation of a poset of size can contain cycles. In this paper we are interested in obtaining minimum cutting levels such that the crisp relation obtained from the mutual rank probability relation by setting to its elements smaller than or equal to , and to its other elements, is free from cycles of length . In a first part, theoretical upper bounds for are derived using known transitivity properties of the mutual rank probability relation. Next, we experimentally obtain minimum cutting levels for posets of size . 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 is computed. To accomplish this, a family of posets is used that is inspired by the experimentally obtained -element poset requiring the highest cutting level to avoid cycles of length