# The lattice Burnside rings

## TL;DR

This paper introduces the concept of lattice Burnside rings for finite groups, exploring their structure, units, and idempotents, and establishing connections with slice and section Burnside rings.

## Contribution

It defines lattice Burnside rings, relates them to existing Burnside ring variants, and analyzes their algebraic properties and structure.

## Key findings

- Lattice Burnside rings are isomorphic to abstract Burnside rings.
- The structure of units and primitive idempotents in lattice Burnside rings is characterized.
- Connections between lattice, slice, and section Burnside rings are established.

## Abstract

We introduce the concept of lattice Burnside ring for a finite group G associated to a family of nonempty sublattices of a finite G-lattice assigned to subgroups of G. The slice Burnside ring introduced by S. Bouc is isomorphic to a lattice Burnside ring. Any lattice Burnside ring is isomorphic to an abstract Burnside ring. The ring structure of a lattice Burnside ring is explored on the basis of the fundamental theorem of abstract Burnside rings. We study the units and the primitive idempotents of a lattice Burnside ring. There are certain rings called partial lattice Burnside rings. Any partial lattice Burnside ring, which is isomorphic to an abstract Burnside ring, consists of elements of a lattice Burnside ring; it is not necessarily a subring. The section Burnside ring introduced by S. Bouc is isomorphic to a partial lattice Burnside ring.