Shelah's pcf theory describes a certain structure which must exist if $\aleph _{\omega }$ is strong limit and $2^{\aleph _\omega }>\aleph _{\omega _1}$ holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that such partially ordered sets exist.
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.
Belief functions can be taken as an alternative to the classical probability theory, as a generalization of this theory, but also as a non-traditional and sophisticated application of the probability theory. In this paper we abandon the idea of numerically quantified degrees of belief in favour of the case when belief functions take their vahies in partially ordered sets, perhaps enriched to lower or upper semilattices. Such structures seern to be the most general ones to which reasonable and nontrivial parts of the theory of belief functions can be extended and generalized.
Belief functions can be taken as an alternative to the classical probability theory, as a generalization of this theory, but also as a non-traditional and sophisticated application of the probability theory. In this paper we abandon the idea of nnmerically quantified degrees of belief in favour of the case when belief functions take their values in partially ordered sets, perhaps enriched to lower or upper semilattices. Such structures seern to be the most general ones to which reasoriable and nontrivial parts of the theory of belief functions can be extended and generalized.