Probability Distributions on Partially Ordered Sets and Positive Semigroups

The algebraic structure ([0,∞), +, ≤), together with Lebesgue measure (and the usual topology), is the standard representation of time for a variety of probabilistic models. These include reliability theory and the aging of lifetime distributions, renewal processes, and Markov processes, to name just a few. The purpose of this monograph is to study probability models in which the standard representation of time is generalized to a partially ordered set (S, ) and to a natural structure (S, ·, ) known as a positive semigroup. The partial order replaces the ordinary order ≤, and in the case of a positive semigroup, the operation · replaces the standard translation operator +. A reference measure λ on S replaces Lebesgue measure, and in the case of a positive semigroup, we assume that this measure is left-invariant.