Mobius monoids and their connection to inverse monoids
Möbius monoids are Möbius categories in the sense of Leroux having a single object. The paper explores several properties of Möbius monoids and inverse monoids as well as the links between them. The right cancellative, left rigid Möbius monoids have a special place in our study. The paper has two goals: first, an examination of Möbius functions and Möbius inversion under suitable conditions, and second, a development of special inverse monoids and inverse submonoids which arise from Möbius monoids and Möbius categories. The prototype of this development is the polycyclic monoid, more precisely the free monoid as a Möbius monoid. This leads us to the generalization of some significant results on polycyclic monoids as Meakin–Sapir’s result involving self-conjugate inverse submonoids, or as Jones-Lawson’s results involving gauge inverse submonoids of the polycyclic monoids. Regarding Möbius functions we create links between several types of these functions.