Decomposition Theory

Abstract

In this chapter the main results on decomposition theory are revisited using the tools of species and set operads. We begin by giving the definition of module domain operads, and by introducing the sub-classes of partitive and weakly partitive operads. If an operad is either partitive or weakly partitive, each of its structures has a unique factorization into prime factors. We then introduce the amalgam operation, based upon which we can construct operads where a unique factorization is still valid, but whose amalgam factors are not weakly partitive, thus widening the spectrum of possibilities of unique factorizable structures.