Lattice Burnside rings
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Algebra Universalis
Lattice Burnside rings Fumihito Oda, Yugen Takegahara and Tomoyuki Yoshida Abstract. Given a finite group G and a finite G-lattice L , we introduce the concept of lattice Burnside ring associated to a family of nonempty sublattices LH of L for H ≤ 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. Mathematics Subject Classification. 19A22, 16U40, 16U60. Keywords. Abstract Burnside ring, Finite lattice, Monoid, Prime spectrum, Primitive idempotent, Unit group.
1. Introduction Let G be a finite group, and let L be a finite G-lattice, that is, L is a finite lattice on which G acts and the binary relation ≤ is invariant under the action of G. The purpose of this paper is to introduce the concept of lattice Burnside ring associated to a family of nonempty sublattices LH of L for H ≤ G, together with its ring structure. The slice Burnside ring of G introduced by Bouc [5], which arises from morphisms of finite G-sets, inspired us to study lattice Burnside rings. In Section 2, we first introduce the concept of monoid functor M = (M, con, res) assigning each H ≤ G a monoid M (H). Any monoid functor M = (M, con, res) corresponds to a certain additive contravariant functor Presented by F. Wehrung. This work was supported by JSPS KAKENHI Grant Number JP19K03436 and JP19K03457. 0123456789().: V,-vol
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F. Oda, Y. Takegahara and T. Yoshida
Algebra Univers.
M from the category of finite left G-sets to the category of monoids (see Equations (2.1) and (2.2)). Given a monoid functor M = (M, con, res), the M -Burnside ring ΩM (G) is the F-Burnside ring AF (G) of G-sets over G/G with F = M (cf. [11,13]). Any finite lattice is considered as a monoid with the binary operation given by ‘meet’, so that L is a finite monoid. Given a family of nonempty sublattices LH for H ≤ G such that the conditions (1)–(4) in Proposition 3.2 are fulfilled, which is said to be admissible, we define a monoid functor ML = (ML , con, res) with ML (H) = LH for each H ≤ G, and call ΩML (G) the lattice Burnside ring of G on the admissible family { LH | H ≤ G } of nonempty sublattices of L . There exists a monoid functor CL = (CL , con, res) assigning each H ≤ G the set of H-invariants in L , and the lattice Burnside ring ΩCL (G) is isomorphic to the crossed Burnside ring associated to L . We consider the set S := S (G) of subg
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