Centers of monoids

Main definitions

• Submonoid.center: the center of a monoid
• AddSubmonoid.center: the center of an additive monoid

We provide Subgroup.center, AddSubgroup.center, Subsemiring.center, and Subring.center in other files.

theorem AddSubmonoid.center.proof_2 (M : Type u_1) [AddMonoid M] : 0 ∈ Set.addCenter M

def AddSubmonoid.center (M : Type u_1) [AddMonoid M] : AddSubmonoid M

The center of a monoid M is the set of elements that commute with everything in M

Equations

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

theorem AddSubmonoid.center.proof_1 (M : Type u_1) [AddMonoid M] : ∀ {a b : M}, a ∈ Set.addCenter M → b ∈ Set.addCenter M → a + b ∈ Set.addCenter M

def Submonoid.center (M : Type u_1) [Monoid M] : Submonoid M

The center of a monoid M is the set of elements that commute with everything in M

Equations

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

theorem AddSubmonoid.coe_center (M : Type u_1) [AddMonoid M] : ↑(AddSubmonoid.center M) = Set.addCenter M

theorem Submonoid.coe_center (M : Type u_1) [Monoid M] : ↑(Submonoid.center M) = Set.center M

@[simp]

theorem AddSubmonoid.center_toAddSubsemigroup (M : Type u_1) [AddMonoid M] : (AddSubmonoid.center M).toAddSubsemigroup = AddSubsemigroup.center M

@[simp]

theorem Submonoid.center_toSubsemigroup (M : Type u_1) [Monoid M] : (Submonoid.center M).toSubsemigroup = Subsemigroup.center M

theorem AddSubmonoid.mem_center_iff {M : Type u_1} [AddMonoid M] {z : M} : z ∈ AddSubmonoid.center M ↔ ∀ (g : M), g + z = z + g

theorem Submonoid.mem_center_iff {M : Type u_1} [Monoid M] {z : M} : z ∈ Submonoid.center M ↔ ∀ (g : M), g * z = z * g

instance AddSubmonoid.decidableMemCenter {M : Type u_1} [AddMonoid M] (a : M) [Decidable (∀ (b : M), b + a = a + b)] : Decidable (a ∈ AddSubmonoid.center M)

Equations

• AddSubmonoid.decidableMemCenter a = decidable_of_iff' (∀ (g : M), g + a = a + g) (_ : a ∈ AddSubmonoid.center M ↔ ∀ (g : M), g + a = a + g)

instance Submonoid.decidableMemCenter {M : Type u_1} [Monoid M] (a : M) [Decidable (∀ (b : M), b * a = a * b)] : Decidable (a ∈ Submonoid.center M)

Equations

• Submonoid.decidableMemCenter a = decidable_of_iff' (∀ (g : M), g * a = a * g) (_ : a ∈ Submonoid.center M ↔ ∀ (g : M), g * a = a * g)

instance Submonoid.center.commMonoid {M : Type u_1} [Monoid M] : CommMonoid { x // x ∈ Submonoid.center M }

The center of a monoid is commutative.

Equations

• Submonoid.center.commMonoid = let src := Submonoid.toMonoid (Submonoid.center M); CommMonoid.mk (_ : ∀ (x b : { x // x ∈ Submonoid.center M }), x * b = b * x)

instance Submonoid.center.smulCommClass_left {M : Type u_1} [Monoid M] : SMulCommClass { x // x ∈ Submonoid.center M } M M

The center of a monoid acts commutatively on that monoid.

Equations

• @Submonoid.center.smulCommClass_left = @Submonoid.center.smulCommClass_left.proof_1

instance Submonoid.center.smulCommClass_right {M : Type u_1} [Monoid M] : SMulCommClass M { x // x ∈ Submonoid.center M } M

The center of a monoid acts commutatively on that monoid.

Equations

• @Submonoid.center.smulCommClass_right = @Submonoid.center.smulCommClass_right.proof_1

Note that smulCommClass (center M) (center M) M is already implied by Submonoid.smulCommClass_right

@[simp]

theorem Submonoid.center_eq_top (M : Type u_1) [CommMonoid M] : Submonoid.center M = ⊤

theorem addUnitsCenterToCenterAddUnits.proof_2 (M : Type u_1) [AddMonoid M] (u : AddUnits { x // x ∈ AddSubmonoid.center M }) (r : AddUnits M) : r + ↑(AddUnits.map (AddSubmonoid.subtype (AddSubmonoid.center M))) u = ↑(AddUnits.map (AddSubmonoid.subtype (AddSubmonoid.center M))) u + r

def addUnitsCenterToCenterAddUnits (M : Type u_1) [AddMonoid M] : AddUnits { x // x ∈ AddSubmonoid.center M } →+ { x // x ∈ AddSubmonoid.center (AddUnits M) }

For an additive monoid, the units of the center inject into the center of the units.

Equations

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

theorem addUnitsCenterToCenterAddUnits.proof_1 (M : Type u_1) [AddMonoid M] : AddSubmonoidClass (AddSubmonoid (AddUnits M)) (AddUnits M)

@[simp]

theorem val_addUnitsCenterToCenterAddUnits_apply_coe (M : Type u_1) [AddMonoid M] (n : AddUnits { x // x ∈ AddSubmonoid.center M }) : ↑↑(↑(addUnitsCenterToCenterAddUnits M) n) = ↑↑n

@[simp]

theorem val_unitsCenterToCenterUnits_apply_coe (M : Type u_1) [Monoid M] (n : { x // x ∈ Submonoid.center M }ˣ) : ↑↑(↑(unitsCenterToCenterUnits M) n) = ↑↑n

def unitsCenterToCenterUnits (M : Type u_1) [Monoid M] : { x // x ∈ Submonoid.center M }ˣ →* { x // x ∈ Submonoid.center Mˣ }

For a monoid, the units of the center inject into the center of the units. This is not an equivalence in general; one case when it is is for groups with zero, which is covered in centerUnitsEquivUnitsCenter.

Equations

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

theorem addUnitsCenterToCenterAddUnits_injective (M : Type u_1) [AddMonoid M] : Function.Injective ↑(addUnitsCenterToCenterAddUnits M)

theorem unitsCenterToCenterUnits_injective (M : Type u_1) [Monoid M] : Function.Injective ↑(unitsCenterToCenterUnits M)