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.

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' := ⋯ }

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' := ⋯ }

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

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

@[simp]

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

@[simp]

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

@[reducible, inline]

abbrev Submonoid.center.commMonoid' {M : Type u_1} [MulOneClass M] : CommMonoid ↥(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

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

@[reducible, inline]

abbrev AddSubmonoid.center.addCommMonoid' {M : Type u_1} [AddZeroClass M] : AddCommMonoid ↥(center M)

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

Equations

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

instance Submonoid.center.commMonoid {M : Type u_1} [Monoid M] : CommMonoid ↥(center M)

The center of a monoid is commutative.

Equations

• Submonoid.center.commMonoid = { toMonoid := (Submonoid.center M).toMonoid, mul_comm := ⋯ }

instance AddSubmonoid.center.addCommMonoid {M : Type u_1} [AddMonoid M] : AddCommMonoid ↥(center M)

Equations

• AddSubmonoid.center.addCommMonoid = { toAddMonoid := (AddSubmonoid.center M).toAddMonoid, add_comm := ⋯ }

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

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

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

Equations

• Submonoid.decidableMemCenter a = decidable_of_iff' (∀ (g : M), g * a = a * g) ⋯

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

Equations

• AddSubmonoid.decidableMemCenter a = decidable_of_iff' (∀ (g : M), g + a = a + g) ⋯

instance Submonoid.center.smulCommClass_left {M : Type u_1} [Monoid M] : SMulCommClass (↥(center M)) M M

The center of a monoid acts commutatively on that monoid.

instance Submonoid.center.smulCommClass_right {M : Type u_1} [Monoid M] : SMulCommClass M (↥(center M)) M

The center of a monoid acts commutatively on that monoid.

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] : center M = ⊤

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

def addUnitsCenterToCenterAddUnits (M : Type u_1) [AddMonoid M] : AddUnits ↥(AddSubmonoid.center M) →+ ↥(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)) ⋯

@[simp]

theorem val_unitsCenterToCenterUnits_apply_coe (M : Type u_1) [Monoid M] (n : (↥(Submonoid.center M))ˣ) : ↑↑((unitsCenterToCenterUnits M) n) = ↑↑n

@[simp]

theorem val_addUnitsCenterToCenterAddUnits_apply_coe (M : Type u_1) [AddMonoid M] (n : AddUnits ↥(AddSubmonoid.center M)) : ↑↑((addUnitsCenterToCenterAddUnits M) n) = ↑↑n

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

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

theorem MulEquivClass.apply_mem_center {M : Type u_3} {N : Type u_1} {F : Type u_2} [EquivLike F M N] [Mul M] [Mul N] [MulEquivClass F M N] (e : F) {x : M} (hx : x ∈ Set.center M) : e x ∈ Set.center N

theorem AddEquivClass.apply_mem_center {M : Type u_3} {N : Type u_1} {F : Type u_2} [EquivLike F M N] [Add M] [Add N] [AddEquivClass F M N] (e : F) {x : M} (hx : x ∈ Set.addCenter M) : e x ∈ Set.addCenter N

theorem MulEquivClass.apply_mem_center_iff {M : Type u_3} {N : Type u_1} {F : Type u_2} [EquivLike F M N] [Mul M] [Mul N] [MulEquivClass F M N] (e : F) {x : M} : e x ∈ Set.center N ↔ x ∈ Set.center M

theorem AddEquivClass.apply_mem_center_iff {M : Type u_3} {N : Type u_1} {F : Type u_2} [EquivLike F M N] [Add M] [Add N] [AddEquivClass F M N] (e : F) {x : M} : e x ∈ Set.addCenter N ↔ x ∈ Set.addCenter M

def Subsemigroup.centerCongr {M : Type u_2} {N : Type u_1} [Mul M] [Mul N] (e : M ≃* N) : ↥(center M) ≃* ↥(center N)

The center of isomorphic magmas are isomorphic.

Equations

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

def AddSubsemigroup.centerCongr {M : Type u_2} {N : Type u_1} [Add M] [Add N] (e : M ≃+ N) : ↥(center M) ≃+ ↥(center N)

The center of isomorphic additive magmas are isomorphic.

Equations

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

@[simp]

theorem Subsemigroup.centerCongr_symm_apply_coe {M : Type u_2} {N : Type u_1} [Mul M] [Mul N] (e : M ≃* N) (s : ↥(center N)) : ↑((centerCongr e).symm s) = e.symm ↑s

@[simp]

theorem AddSubsemigroup.centerCongr_symm_apply_coe {M : Type u_2} {N : Type u_1} [Add M] [Add N] (e : M ≃+ N) (s : ↥(center N)) : ↑((centerCongr e).symm s) = e.symm ↑s

@[simp]

theorem AddSubsemigroup.centerCongr_apply_coe {M : Type u_2} {N : Type u_1} [Add M] [Add N] (e : M ≃+ N) (r : ↥(center M)) : ↑((centerCongr e) r) = e ↑r

@[simp]

theorem Subsemigroup.centerCongr_apply_coe {M : Type u_2} {N : Type u_1} [Mul M] [Mul N] (e : M ≃* N) (r : ↥(center M)) : ↑((centerCongr e) r) = e ↑r

def Submonoid.centerCongr {M : Type u_2} {N : Type u_1} [MulOneClass M] [MulOneClass N] (e : M ≃* N) : ↥(center M) ≃* ↥(center N)

The center of isomorphic monoids are isomorphic.

Equations

• Submonoid.centerCongr e = Subsemigroup.centerCongr e

def AddSubmonoid.centerCongr {M : Type u_2} {N : Type u_1} [AddZeroClass M] [AddZeroClass N] (e : M ≃+ N) : ↥(center M) ≃+ ↥(center N)

The center of isomorphic additive monoids are isomorphic.

Equations

• AddSubmonoid.centerCongr e = AddSubsemigroup.centerCongr e

@[simp]

theorem AddSubmonoid.centerCongr_apply_coe {M : Type u_2} {N : Type u_1} [AddZeroClass M] [AddZeroClass N] (e : M ≃+ N) (r : ↥(AddSubsemigroup.center M)) : ↑((centerCongr e) r) = e ↑r

@[simp]

theorem Submonoid.centerCongr_symm_apply_coe {M : Type u_2} {N : Type u_1} [MulOneClass M] [MulOneClass N] (e : M ≃* N) (s : ↥(Subsemigroup.center N)) : ↑((centerCongr e).symm s) = e.symm ↑s

@[simp]

theorem AddSubmonoid.centerCongr_symm_apply_coe {M : Type u_2} {N : Type u_1} [AddZeroClass M] [AddZeroClass N] (e : M ≃+ N) (s : ↥(AddSubsemigroup.center N)) : ↑((centerCongr e).symm s) = e.symm ↑s

@[simp]

theorem Submonoid.centerCongr_apply_coe {M : Type u_2} {N : Type u_1} [MulOneClass M] [MulOneClass N] (e : M ≃* N) (r : ↥(Subsemigroup.center M)) : ↑((centerCongr e) r) = e ↑r

theorem MulOpposite.op_mem_center_iff {M : Type u_2} [Mul M] {x : M} : op x ∈ Set.center Mᵐᵒᵖ ↔ x ∈ Set.center M

theorem AddOpposite.op_mem_center_iff {M : Type u_2} [Add M] {x : M} : op x ∈ Set.addCenter Mᵃᵒᵖ ↔ x ∈ Set.addCenter M

theorem MulOpposite.unop_mem_center_iff {M : Type u_2} [Mul M] {x : Mᵐᵒᵖ} : unop x ∈ Set.center M ↔ x ∈ Set.center Mᵐᵒᵖ

theorem AddOpposite.unop_mem_center_iff {M : Type u_2} [Add M] {x : Mᵃᵒᵖ} : unop x ∈ Set.addCenter M ↔ x ∈ Set.addCenter Mᵃᵒᵖ

def Subsemigroup.centerToMulOpposite {M : Type u_2} [Mul M] : ↥(center M) ≃* ↥(center Mᵐᵒᵖ)

The center of a magma is isomorphic to the center of its opposite.

Equations

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

def AddSubsemigroup.centerToAddOpposite {M : Type u_2} [Add M] : ↥(center M) ≃+ ↥(center Mᵃᵒᵖ)

The center of an additive magma is isomorphic to the center of its opposite.

Equations

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

@[simp]

theorem Subsemigroup.centerToMulOpposite_apply_coe {M : Type u_2} [Mul M] (r : ↥(center M)) : ↑(centerToMulOpposite r) = MulOpposite.op ↑r

@[simp]

theorem AddSubsemigroup.centerToAddOpposite_symm_apply_coe {M : Type u_2} [Add M] (r : ↥(center Mᵃᵒᵖ)) : ↑(centerToAddOpposite.symm r) = AddOpposite.unop ↑r

@[simp]

theorem AddSubsemigroup.centerToAddOpposite_apply_coe {M : Type u_2} [Add M] (r : ↥(center M)) : ↑(centerToAddOpposite r) = AddOpposite.op ↑r

@[simp]

theorem Subsemigroup.centerToMulOpposite_symm_apply_coe {M : Type u_2} [Mul M] (r : ↥(center Mᵐᵒᵖ)) : ↑(centerToMulOpposite.symm r) = MulOpposite.unop ↑r

def Submonoid.centerToMulOpposite {M : Type u_2} [MulOneClass M] : ↥(center M) ≃* ↥(center Mᵐᵒᵖ)

The center of a monoid is isomorphic to the center of its opposite.

Equations

• Submonoid.centerToMulOpposite = Subsemigroup.centerToMulOpposite

def AddSubmonoid.centerToAddOpposite {M : Type u_2} [AddZeroClass M] : ↥(center M) ≃+ ↥(center Mᵃᵒᵖ)

The center of an additive monoid is isomorphic to the center of its opposite.

Equations

• AddSubmonoid.centerToAddOpposite = AddSubsemigroup.centerToAddOpposite

@[simp]

theorem AddSubmonoid.centerToAddOpposite_symm_apply_coe {M : Type u_2} [AddZeroClass M] (r : ↥(AddSubsemigroup.center Mᵃᵒᵖ)) : ↑(centerToAddOpposite.symm r) = AddOpposite.unop ↑r

@[simp]

theorem Submonoid.centerToMulOpposite_apply_coe {M : Type u_2} [MulOneClass M] (r : ↥(Subsemigroup.center M)) : ↑(centerToMulOpposite r) = MulOpposite.op ↑r

@[simp]

theorem Submonoid.centerToMulOpposite_symm_apply_coe {M : Type u_2} [MulOneClass M] (r : ↥(Subsemigroup.center Mᵐᵒᵖ)) : ↑(centerToMulOpposite.symm r) = MulOpposite.unop ↑r

@[simp]

theorem AddSubmonoid.centerToAddOpposite_apply_coe {M : Type u_2} [AddZeroClass M] (r : ↥(AddSubsemigroup.center M)) : ↑(centerToAddOpposite r) = AddOpposite.op ↑r