A Burnside ring for monoids

Abstract

We define the Burnside ring of a monoid, generalizing the construction for groups. After giving foundational definitions, we describe the monoid-theoretic correlates of orbits and their automorphisms, then prove a structure theorem for a broad class of monoids that allows us to write the Burnside ring as a direct product of Burnside rings of groups. Finally, we define a monoid-theoretic correlate of the table of marks, and show that the Burnside algebra over $Q$ is semisimple.