Fixing submonoid, fixing subgroup of an action

In the presence of an action of a monoid or a group, this file defines the fixing submonoid or the fixing subgroup, and relates it to the set of fixed points via a Galois connection.

Main definitions

• fixingSubmonoid M s : in the presence of MulAction M α (with Monoid M) it is the Submonoid M consisting of elements which fix s : Set α pointwise.

• fixingSubmonoid_fixedPoints_gc M α is the GaloisConnection that relates fixingSubmonoid with fixedPoints.

• fixingSubgroup M s : in the presence of MulAction M α (with Group M) it is the Subgroup M consisting of elements which fix s : Set α pointwise.

• fixingSubgroup_fixedPoints_gc M α is the GaloisConnection that relates fixingSubgroup with fixedPoints.

TODO :

• Maybe other lemmas are useful

• Treat semigroups ?

def fixingAddSubmonoid (M : Type u_1) {α : Type u_2} [AddMonoid M] [AddAction M α] (s : Set α) : AddSubmonoid M

The additive submonoid fixing a set under an AddAction.

Equations

• One or more equations did not get rendered due to their size.

theorem fixingAddSubmonoid.proof_2 (M : Type u_2) {α : Type u_1} [AddMonoid M] [AddAction M α] (s : Set α) : ∀ (x : ↑s), 0 +ᵥ ↑x = ↑x

theorem fixingAddSubmonoid.proof_1 (M : Type u_1) {α : Type u_2} [AddMonoid M] [AddAction M α] (s : Set α) {x : M} {y : M} (hx : x ∈ {ϕ | ∀ (x : ↑s), ϕ +ᵥ ↑x = ↑x}) (hy : y ∈ {ϕ | ∀ (x : ↑s), ϕ +ᵥ ↑x = ↑x}) (z : ↑s) : x + y +ᵥ ↑z = ↑z

def fixingSubmonoid (M : Type u_1) {α : Type u_2} [Monoid M] [MulAction M α] (s : Set α) : Submonoid M

The submonoid fixing a set under a MulAction.

Equations

• One or more equations did not get rendered due to their size.

theorem mem_fixingSubmonoid_iff (M : Type u_1) {α : Type u_2} [Monoid M] [MulAction M α] {s : Set α} {m : M} : m ∈ fixingSubmonoid M s ↔ ∀ (y : α), y ∈ s → m • y = y

theorem fixingSubmonoid_fixedPoints_gc (M : Type u_1) (α : Type u_2) [Monoid M] [MulAction M α] : GaloisConnection (↑OrderDual.toDual ∘ fixingSubmonoid M) ((fun P => MulAction.fixedPoints { x // x ∈ P } α) ∘ ↑OrderDual.ofDual)

The Galois connection between fixing submonoids and fixed points of a monoid action

theorem fixingSubmonoid_antitone (M : Type u_1) (α : Type u_2) [Monoid M] [MulAction M α] : Antitone fun s => fixingSubmonoid M s

theorem fixedPoints_antitone (M : Type u_1) (α : Type u_2) [Monoid M] [MulAction M α] : Antitone fun P => MulAction.fixedPoints { x // x ∈ P } α

theorem fixingSubmonoid_union (M : Type u_1) (α : Type u_2) [Monoid M] [MulAction M α] {s : Set α} {t : Set α} : fixingSubmonoid M (s ∪ t) = fixingSubmonoid M s ⊓ fixingSubmonoid M t

Fixing submonoid of union is intersection

theorem fixingSubmonoid_iUnion (M : Type u_1) (α : Type u_2) [Monoid M] [MulAction M α] {ι : Sort u_3} {s : ι → Set α} : fixingSubmonoid M (⋃ (i : ι), s i) = ⨅ (i : ι), fixingSubmonoid M (s i)

Fixing submonoid of iUnion is intersection

theorem fixedPoints_submonoid_sup (M : Type u_1) (α : Type u_2) [Monoid M] [MulAction M α] {P : Submonoid M} {Q : Submonoid M} : MulAction.fixedPoints { x // x ∈ P ⊔ Q } α = MulAction.fixedPoints { x // x ∈ P } α ∩ MulAction.fixedPoints { x // x ∈ Q } α

Fixed points of sup of submonoids is intersection

theorem fixedPoints_submonoid_iSup (M : Type u_1) (α : Type u_2) [Monoid M] [MulAction M α] {ι : Sort u_3} {P : ι → Submonoid M} : MulAction.fixedPoints { x // x ∈ iSup P } α = ⋂ (i : ι), MulAction.fixedPoints { x // x ∈ P i } α

Fixed points of iSup of submonoids is intersection

theorem fixingAddSubgroup.proof_2 (M : Type u_1) {α : Type u_2} [AddGroup M] [AddAction M α] (s : Set α) : ∀ {x : M}, x ∈ { toAddSubsemigroup := (fixingAddSubmonoid M s).toAddSubsemigroup, zero_mem' := (_ : 0 ∈ (fixingAddSubmonoid M s).toAddSubsemigroup.carrier) }.toAddSubsemigroup.carrier → ∀ (z : ↑s), -x +ᵥ ↑z = ↑z

def fixingAddSubgroup (M : Type u_1) {α : Type u_2} [AddGroup M] [AddAction M α] (s : Set α) : AddSubgroup M

The additive subgroup fixing a set under an AddAction.

Equations

• One or more equations did not get rendered due to their size.

theorem fixingAddSubgroup.proof_1 (M : Type u_1) {α : Type u_2} [AddGroup M] [AddAction M α] (s : Set α) : 0 ∈ (fixingAddSubmonoid M s).toAddSubsemigroup.carrier

def fixingSubgroup (M : Type u_1) {α : Type u_2} [Group M] [MulAction M α] (s : Set α) : Subgroup M

The subgroup fixing a set under a MulAction.

Equations

• One or more equations did not get rendered due to their size.

theorem mem_fixingSubgroup_iff (M : Type u_1) {α : Type u_2} [Group M] [MulAction M α] {s : Set α} {m : M} : m ∈ fixingSubgroup M s ↔ ∀ (y : α), y ∈ s → m • y = y

theorem fixingSubgroup_fixedPoints_gc (M : Type u_1) (α : Type u_2) [Group M] [MulAction M α] : GaloisConnection (↑OrderDual.toDual ∘ fixingSubgroup M) ((fun P => MulAction.fixedPoints { x // x ∈ P } α) ∘ ↑OrderDual.ofDual)

The Galois connection between fixing subgroups and fixed points of a group action

theorem fixingSubgroup_antitone (M : Type u_1) (α : Type u_2) [Group M] [MulAction M α] : Antitone (fixingSubgroup M)

theorem fixedPoints_subgroup_antitone (M : Type u_1) (α : Type u_2) [Group M] [MulAction M α] : Antitone fun P => MulAction.fixedPoints { x // x ∈ P } α

theorem fixingSubgroup_union (M : Type u_1) (α : Type u_2) [Group M] [MulAction M α] {s : Set α} {t : Set α} : fixingSubgroup M (s ∪ t) = fixingSubgroup M s ⊓ fixingSubgroup M t

Fixing subgroup of union is intersection

theorem fixingSubgroup_iUnion (M : Type u_1) (α : Type u_2) [Group M] [MulAction M α] {ι : Sort u_3} {s : ι → Set α} : fixingSubgroup M (⋃ (i : ι), s i) = ⨅ (i : ι), fixingSubgroup M (s i)

Fixing subgroup of iUnion is intersection

theorem fixedPoints_subgroup_sup (M : Type u_1) (α : Type u_2) [Group M] [MulAction M α] {P : Subgroup M} {Q : Subgroup M} : MulAction.fixedPoints { x // x ∈ P ⊔ Q } α = MulAction.fixedPoints { x // x ∈ P } α ∩ MulAction.fixedPoints { x // x ∈ Q } α

Fixed points of sup of subgroups is intersection

theorem fixedPoints_subgroup_iSup (M : Type u_1) (α : Type u_2) [Group M] [MulAction M α] {ι : Sort u_3} {P : ι → Subgroup M} : MulAction.fixedPoints { x // x ∈ iSup P } α = ⋂ (i : ι), MulAction.fixedPoints { x // x ∈ P i } α

Fixed points of iSup of subgroups is intersection