Möbius functions on locally finite posets

Abstract

In the recent years an important mathematical structure, the partially ordered sets have provided a framework for unifying a large part of combinatorial analysis. The generalized Mobius inversion is formulated in the language of incidence algebra over locally finite partially ordered sets, initiated by Gian-Carlo Rota who made the Mobius function. and hence its associated zeta function, a central unifying concept in combinatorics. We shall present the elements of this generalization and a few of its applications in this thesis. Furthermore, in an attempt to seek an efficient formulation of Rota's classical formula that describes the relationship between the Mobius functions of two partially ordered sets P and Q related by a Galois connection, we interpret this connection in the language of module theory. In the process, an I, -I, - bimodule Mf over the incidence algebras I, and I,- is constructed with respect to a class of monotone maps and thus another proof of the Rota's formula is obtained.