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## Über dieses Buch

This monograph has two main objectives. The first one is to give a self-contained exposition of the relevant facts about set operads, in the context of combinatorial species and its operations. This approach has various advantages: one of them is that the definition of combinatorial operations on species, product, sum, substitution and derivative, are simple and natural. They were designed as the set theoretical counterparts of the homonym operations on exponential generating functions, giving an immediate insight on the combinatorial meaning of them. The second objective is more ambitious. Before formulating it, authors present a brief historic account on the sources of decomposition theory. For more than forty years decompositions of discrete structures have been studied in different branches of discrete mathematics: combinatorial optimization, network and graph theory, switching design or boolean functions, simple multi-person games and clutters, etc.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
In this chapter the main ideas involved in the notion of a set operad are explained by studying the example of parenthesized injective words, and a historical account on set operads is provided. Finally, we state the main objectives of this monograph.
Miguel A. Méndez

### 2. Preliminaries on Species and Set Operads

Abstract
The formal definition of a species and its exponential generating function are given. Based on the intuitive definition of the operation of substitution of species we give an informal definition of a set operad.
Miguel A. Méndez

### 3. 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.
Miguel A. Méndez

### 4. 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.
Miguel A. Méndez

Abstract
A rigid finite structure is one without automorphisms other than the identity. Structures constructed over totally ordered sets are rigid. Thus, the natural context for the definition of rigid operads is the species on linear orders or L-species. In the context of $$\mathcal{L}$$-species there are two different substitution operations; ordinal and shuffle. These operations lead to two kinds of operads, respectively non-symmetric and shuffle.
Miguel A. Méndez

### 6. Posets from Cancellative Operads and Koszul Duality

Abstract
A species $${\mathcal{G}}$$ is said to be homogeneous concentrated in m, if $${\mathcal{G}}[k]=\emptyset$$ for $$k\neq m$$. Based on previous work of Fresse [Fre04], B. Vallette [Val07] proved that a quadratic cancellative operad generated by a homogeneous species is Koszul if and only if the maximal intervals of the associated posets P Q are Cohen–Macaulay. In this chapter, we give an account of Vallete’s results and generalize his criterion for Koszulness. We show that the homogeneity assumption is not necessary. The result is still valid for any quadratic cancellative operad.
Miguel A. Méndez

### Backmatter

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