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_1 (M : Type u_1) [AddZeroClass M] : ∀ {a b : M}, a ∈ Set.addCenter M → b ∈ Set.addCenter M → a + b ∈ Set.addCenter M

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

The center of an addition with zero M is the set of elements that commute with everything in M

Equations

• AddSubmonoid.center M = { carrier := Set.addCenter M, add_mem' := ⋯, zero_mem' := ⋯ }

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

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

Equations

• Submonoid.center M = { carrier := Set.center M, mul_mem' := ⋯, one_mem' := ⋯ }

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

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

@[simp]

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

@[simp]

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

theorem AddSubmonoid.center.addCommMonoid'.proof_5 {M : Type u_1} [AddZeroClass M] : ∀ (n : ℕ) (x : { x : M // x ∈ AddSubmonoid.center M }), nsmulRec (n + 1) x = nsmulRec (n + 1) x

theorem AddSubmonoid.center.addCommMonoid'.proof_2 {M : Type u_1} [AddZeroClass M] (a : { x : M // x ∈ AddSubmonoid.center M }) : 0 + a = a

theorem AddSubmonoid.center.addCommMonoid'.proof_1 {M : Type u_1} [AddZeroClass M] (a : { x : M // x ∈ AddSubsemigroup.center M }) (b : { x : M // x ∈ AddSubsemigroup.center M }) (c : { x : M // x ∈ AddSubsemigroup.center M }) : a + b + c = a + (b + c)

theorem AddSubmonoid.center.addCommMonoid'.proof_3 {M : Type u_1} [AddZeroClass M] (a : { x : M // x ∈ AddSubmonoid.center M }) : a + 0 = a

theorem AddSubmonoid.center.addCommMonoid'.proof_6 {M : Type u_1} [AddZeroClass M] (a : { x : M // x ∈ AddSubsemigroup.center M }) (b : { x : M // x ∈ AddSubsemigroup.center M }) : a + b = b + a

theorem AddSubmonoid.center.addCommMonoid'.proof_4 {M : Type u_1} [AddZeroClass M] : ∀ (x : { x : M // x ∈ AddSubmonoid.center M }), nsmulRec 0 x = nsmulRec 0 x

@[reducible, inline]

abbrev AddSubmonoid.center.addCommMonoid' {M : Type u_1} [AddZeroClass M] : AddCommMonoid { x : M // x ∈ AddSubmonoid.center M }

The center of an addition with zero is commutative and associative.

Equations

• AddSubmonoid.center.addCommMonoid' = AddCommMonoid.mk ⋯

@[reducible, inline]

abbrev Submonoid.center.commMonoid' {M : Type u_1} [MulOneClass M] : CommMonoid { x : M // x ∈ Submonoid.center M }

The center of a multiplication with unit is commutative and associative.

This is not an instance as it forms an non-defeq diamond with Submonoid.toMonoid in the npow field.

Equations

• Submonoid.center.commMonoid' = CommMonoid.mk ⋯

instance AddSubmonoid.center.addCommMonoid {M : Type u_1} [AddMonoid M] : AddCommMonoid { x : M // x ∈ AddSubmonoid.center M }

Equations

• AddSubmonoid.center.addCommMonoid = AddCommMonoid.mk ⋯

theorem AddSubmonoid.center.addCommMonoid.proof_1 {M : Type u_1} [AddMonoid M] (a : { x : M // x ∈ AddSubsemigroup.center M }) (b : { x : M // x ∈ AddSubsemigroup.center M }) : a + b = b + a

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

The center of a monoid is commutative.

Equations

• Submonoid.center.commMonoid = CommMonoid.mk ⋯

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) ⋯

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) ⋯

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

The center of a monoid acts commutatively on that monoid.

Equations

• ⋯ = ⋯

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

The center of a monoid acts commutatively on that monoid.

Equations

• ⋯ = ⋯

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_1 (M : Type u_1) [AddMonoid M] : AddSubmonoidClass (AddSubmonoid (AddUnits M)) (AddUnits M)

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

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

Equations

• addUnitsCenterToCenterAddUnits M = (AddUnits.map (AddSubmonoid.center M).subtype).codRestrict (AddSubmonoid.center (AddUnits M)) ⋯

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

@[simp]

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

@[simp]

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

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

• unitsCenterToCenterUnits M = (Units.map (Submonoid.center M).subtype).codRestrict (Submonoid.center Mˣ) ⋯

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)