Arithmetics of aging distributions: Maximum (Q1320464)

Let \(D^ +\) be the set of all distribution functions of non-negative random variables. For any pair of distributions \(F,G \in D^ +\) let \(F\vee G\) denote their pairwise product. Then \((D^ +,\vee)\) is a semigroup. The author first notes that the classes of IFRA, NBU and NBUE are each a subsemigroup of \((D^ +,\vee)\). He then proceeds to study various properties of these subsemigroups. His studies deal with decompositions of members of these subsemigroups and the identification of the infinitely divisible members of them. The author proves that the sets of irreducible elements of these subsemigroups, as well as the sets of anti-irreducible elements of these subsemigroups, are dense in these subsemigroups. The paper ends with the identification of the prime distributions of these subsemigroups.