Incidence algebras. Overview

Abstract

In his celebrated paper of 1964, "On the foundations of combinatorial theory I: Theory of Möbius Functions" Gian-Carlo Rota defined an incidence algebra as a tool for solving combinatorial problems. Incidence algebra is a specific ring of functions defined on the ordered pairs of a given partially ordered set to a given ring, moreover incidence ring is equipped with a module action by this ring. Möbius function is an element of an incidence algebra, besides with the appropriate choice of the partially ordered set, Möbius function of this incidence algebra coincides with the well-known Möbius function of number theory.

A product of copies of a ring and upper triangular matrices are typical examples of incidence algebras. In the following papers of Rota with his co-authors, and papers of other contemporary authors incidence algebras are investigated as an algebraic object, as a tool in algebraic topology. After a general view of incidence algebras, I will give the connection of incidence algebras to path algebras which are defined on directed graphs. PREREQUISITE: Basic undergraduate algebra will be sufficient.

REFERENCE: Spiegel, O'Donnell Incidence Algebras Pure and Applied Mathematics 206 , Marcel Dekker, Inc., 1997.

SHORT SYLLABUS: This short course will consist of a brief introduction to a mathematical structure called incidence algebras and give connections with algebras over graphs namely path and Leavitt path algebras.

First part will be a survey of partially ordered structures; construction of incidence algebras; incidence subalgebras and applications(eg. formal power series ring, Dirichlet series, Riemann zeta function). Then connections between combinatorics and incidence algebras will be mentioned. (eg. Möbius function)

We will introduce some lattice theory to analyze the algebraic structure of incidence algebras; we determine the ideal structure of incidence algebras.(eg. maximal ideals, prime ideals, radicals).

Finally, we consider other algebraic structures on graphs, ie. path algebras, Leavitt path algebras and the correspondence of incidence and path algebras.

Content of the teaching program: "An overview of some algebras over discrete structures" We will introduce three algebras defined over discrete structures, incidence algebras defined over partially ordered sets, the path algebras defined over directed graphs, and the Leavitt path algebras defined over extended directed graphs.

We will look at the historical development and the interactions between these structures and the connection to analysis via C*-graph algebras.