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.
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This work was supported by JSPS KAKENHI Grant Number JP19K03436 and JP19K03457.
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Oda, F., Takegahara, Y. & Yoshida, T. Lattice Burnside rings. Algebra Univers. 81, 53 (2020). https://doi.org/10.1007/s00012-020-00687-1
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DOI: https://doi.org/10.1007/s00012-020-00687-1