On the cycle-transitivity of the mutual rank probability relation of a poset

The mutual rank probability relation associated with a finite poset is a reciprocal relation expressing the probability that a given element succeeds another one in a random linear extension of that poset. We contribute to the characterization of the transitivity of this mutual rank probability relation, also known as proportional probabilistic transitivity, by situating it between strong stochastic transitivity and moderate product transitivity. The methodology used draws upon the cycle-transitivity framework, which is tailor-made for describing the transitivity of reciprocal relations.