Lattice Burnside rings

Abstract

Given a finite group G and a finite G-lattice $${{\mathscr {L}}}$$ , we introduce the concept of lattice Burnside ring associated to a family of nonempty sublattices $${{\mathscr {L}}}_H$$ of $${{\mathscr {L}}}$$ for $$H\le G$$ . The slice Burnside ring introduced by Bouc is isomorphic to a lattice Burnside ring. Any lattice Burnside ring is an extension of the ordinary Burnside ring and is isomorphic to an abstract Burnside ring. The ring structure of a lattice Burnside ring is explored on the basis of the fundamental theorem for abstract Burnside rings. We explore the unit group, the primitive idempotents, and connected components of the prime spectrum of a lattice Burnside ring. There are certain abstract Burnside rings called partial lattice Burnside rings. Any partial lattice Burnside ring consists of elements of a lattice Burnside ring. The section Burnside ring introduced by Bouc, which is a subring of the slice Burnside ring, is isomorphic to a partial lattice Burnside ring.