Möbius inversion formula for monoids with zero. (Q617859)

The Möbius 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. A `locally finite monoid with zero' is a monoid with zero \(M\) such that for any \(x\in M_0=M-\{0_M\}\) the set \(\{(n,x_1,\dots,x_n):x=x_1\cdots x_n,\;x_i\neq 1_M\}\) is finite [see \textit{P. Cartier} and \textit{D. Foata}, Problèmes combinatoires de commutation et réarrangements. Berlin-Heidelberg-New York: Springer-Verlag (1969; Zbl 0186.30101); \textit{S. Eilenberg}, Automata, languages, and machines. Vol. A. New York-London: Academic Press (1974; Zbl 0317.94045)]. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the total contracted monoid algebra, the augmentation ideal or the star operation on proper series. The main concern is to extend the study of the Möbius function to some monoids with zero, in particular, the so-called Rees quotients of locally finite monoids. The development of a system of algebraic and topological notions gives the possibility for a systematic and rigorous treatment of the Möbius inversion formula for locally finite monoids with zero.