Generalized Numerical Semigroups

Submonoids of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$, thought as a straightforward generalization of numerical semigroups, are considered for the first time in a recent work of G. Failla, C. Peterson and R.Utano. Such monoids are called there \emph{generalized numerical semigroups}, and that work may represent the beginning of a systematic study on such matter. This thesis is a possible starting point of this research. Its aim is to gather all the results obtained during the development of my Ph.D research project in this subject. In this work, first we consider how to characterize the set of generators of a generalized numerical semigroup. Successively we look for some particular classes of generalized numerical semigroups in this new context. We focus in particular on symmetric and pseudo-symmetric generalized numerical semigroups. Moreover we introduce in this general context an important tool, very useful for numerical semigroups, that is the \emph{Ap\'ery set}, and some results about it are provided. Some mentioned questions posed in the mentioned work of G. Failla, C. Peterson and R.Utano are also studied here, in particular we provide: algorithms to manage various features of this subject and implemented in the computer algebra software GAP, using the GAP package \texttt{numericalsgps}; tables with several computational data; the definition of a generalization of a well known conjecture formulated for numerical semigroups, namely \emph{Wilf's conjecture}, which is studied here for some classes of generalized numerical semigtoups introduced.