Generating Functions on idempotent semigroups with application to combinatorial analysis

AbstractIt was shown in earlier work that many combinatorial problems can be treated analytically by considering the algebraic structure of the set on which one is counting. The main tool in this analysis is the “incidence algebra” over a partially ordered set and two of its elements, the zeta function and its inverse the Möbius function. When the partially ordered set is a finite commutative semigroup of idempotents (semilattice) the incidence algebra arises in a natural way when considering the semicharacters of the semigroup (homomorphisms to the complex plane). The zeta function is the semicharacter matrix and hence is multiplicative.In this paper we exploit the multiplicativity of the zeta function to introduce a natural convolution operator over the semigroup. We then define a generating function based on the multiplicative nature of the zeta function. This generating function has the property that it converts convolution to ordinary products. Its inversion is given by the Möbius inversion formula. Many examples and applications are presented

Some Stochastic Inventory Models for Rental Situations

Some stochastic models are analyzed which describe the time fluctuations of the inventory levels of companies which are in the rental business. The stochastic process which describes the fluctuations is shown to be applicable to a wide variety of physical situations. Examples are given, and analyses are made of several types of profit functions which are applicable to rental situations.