Operations on Species and Operads

Abstract

In this chapter operations on species (sum, product, Hadamard product, derivative, and substitution) are introduced by mimicking the analogous operations on exponential generating functions. The operations of substitution and product define two important monoidal categories on species. In this context, operads are defined as monoids in the monoidal category of species with respect to the substitution. Species with a monoidal structure with respect to the operation of product are simply called monoids. As a consequence of the chain rule for species, it is proved that the derivative sends operads into monoids. A construction of families of partially ordered sets from cancellative operads (respectively, cancellative monoids) is obtained. The substitutional (respectively, multiplicative) inverse of the exponential generating function of a cancellative operad (respectively, cancellative monoid) is proved to be the M\"obius generating function of the respective associated family of posets.