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# Generalizations and Extensions of the Möbius Function

## 2.1 Introduction

In this chapter we will survey many generalizations in Number theory of the Möbius function, as well as analogues functions which arose by the extensions of certain divisibility notions or product notions of arithmetical functions. Our study will include also Möbius functions in Group theory, Lattice theory, Partially ordered sets or Arithmetical semigroups. Sometimes references to applications will be pointed out, too. This is a refined and extended version of [164].

((1)) |

((2)) |

However, its arithmetical importance was first recognized by A. F. Möbius [143] in 1832,