This chapter introduces the concept of numerical semigroups. Several properties of the gaps and nongaps of a semigroup are investigated, and the importance of the role played by the Frobenius number (also known as conductor) in the study of symmetric and pseudo-symmetric semigroups is pointed out. A number of results relating FP to telescopic semigroups, the famous Apéry Sets (used by R. Apéry in the study of algebroid planar branches), type sequences in semigroups, complete intersection semigroups, -hyperelliptic semigroups (motivated by the study of Weierstrass semigroups), the Möbius function, and other related concepts are proved.
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