M\"obius inversion formula for monoids with zero

TL;DR

This paper extends the M"obius inversion formula to monoids with zero, exploring algebraic and topological notions, and investigates the relationship between monoids and their Rees quotients.

Contribution

It develops a framework for M"obius functions on monoids with zero, including Rees quotients, and relates these to classical monoid structures.

Findings

Extended M"obius function to monoids with zero

Analyzed Rees quotients of monoids with zero

Established relations between M"obius functions of monoids and their quotients

Abstract

The M\"obius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra of a monoid, the augmentation ideal or the star operation on proper series. The main concern is to extend the study of the M\"obius function to some monoids with zero, i.e., with an absorbing element, in particular the so-called Rees quotients of locally finite monoids. Some relations between the M\"obius functions of a monoid and its Rees quotient are also provided.