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Ideals of some Green biset functors

J. Miguel Calderón

Abstract

In this article, we describe the lattice of ideals of some Green biset functors. We consider Green biset functors which satisfy that each evaluation is a finite dimensional split semisimple commutative algebra and use the idempotents in these evaluations to characterize any ideal of these Green biset functors. For this we will give the definition of M C-group, this definition generalizes that of a B-group, given for the Burnside functor. Given a Green biset functor A, with the above hypotheses, the set of all M C-groups of A has a structure of a poset and we prove that there exists an isomorphism of lattices between the set of ideals of A and the set of upward closed subsets of the M C-groups of A.

Ideals of some Green biset functors

Abstract

In this article, we describe the lattice of ideals of some Green biset functors. We consider Green biset functors which satisfy that each evaluation is a finite dimensional split semisimple commutative algebra and use the idempotents in these evaluations to characterize any ideal of these Green biset functors. For this we will give the definition of M C-group, this definition generalizes that of a B-group, given for the Burnside functor. Given a Green biset functor A, with the above hypotheses, the set of all M C-groups of A has a structure of a poset and we prove that there exists an isomorphism of lattices between the set of ideals of A and the set of upward closed subsets of the M C-groups of A.
Paper Structure (16 sections, 34 theorems, 103 equations)

This paper contains 16 sections, 34 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Green Biset functors
  4. $A$-modules
  5. Examples of Green biset functors
  6. The Burnside functor RB.
  7. Fibered Burnside Functor
  8. Slice Burnside functor
  9. The shifted Functor
  10. Biset Operations on Idempotents
  11. The ideals of the Green biset functor $A$
  12. Examples
  13. The Burnside Functor
  14. Fibered $p$-biset Functor
  15. The slice Burnside functor
  16. ...and 1 more sections

Key Result

Lemma 2.3

The two previous definitions are equivalent. Starting with Definition Defi1, the ring structure of $A(H)$ is given by for $a$ and $b$ in $A(H)$, with the unity given by $A(Inf_1^H)(\xi_A)$. Conversely, starting with Definition green, the product of $A(G) \times A(H) \longrightarrow A(G \times H)$ is given by for $a\in A(G)$ and $b\in A(H)$, with the identity element given by the unity of $A(1)

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Lemma 3 in center
  • Definition 2.4: Definition 8.5.5 in serge-biset
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 64 more