Centers of magmas and semigroups

Main definitions

• Set.center: the center of a magma
• Set.addCenter: the center of an additive magma

See Mathlib.GroupTheory.Subsemigroup.Center for the definition of the center as a subsemigroup:

• Subsemigroup.center: the center of a semigroup
• AddSubsemigroup.center: the center of an additive semigroup

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

structure IsAddCentral {M : Type u_1} [Add M] (z : M) : Prop

Conditions for an element to be additively central

• comm : ∀ (a : M), z + a = a + z

  addition commutes

• left_assoc : ∀ (b c : M), z + (b + c) = z + b + c

  associative property for left addition

• mid_assoc : ∀ (a c : M), a + z + c = a + (z + c)

  middle associative addition property

• right_assoc : ∀ (a b : M), a + b + z = a + (b + z)

  associative property for right addition

structure IsMulCentral {M : Type u_1} [Mul M] (z : M) : Prop

Conditions for an element to be multiplicatively central

• comm : ∀ (a : M), z * a = a * z

  multiplication commutes

• left_assoc : ∀ (b c : M), z * (b * c) = z * b * c

  associative property for left multiplication

• mid_assoc : ∀ (a c : M), a * z * c = a * (z * c)

  middle associative multiplication property

• right_assoc : ∀ (a b : M), a * b * z = a * (b * z)

  associative property for right multiplication

theorem isAddCentral_iff {M : Type u_1} [Add M] (z : M) : IsAddCentral z ↔ (∀ (a : M), z + a = a + z) ∧ (∀ (b c : M), z + (b + c) = z + b + c) ∧ (∀ (a c : M), a + z + c = a + (z + c)) ∧ ∀ (a b : M), a + b + z = a + (b + z)

theorem isMulCentral_iff {M : Type u_1} [Mul M] (z : M) : IsMulCentral z ↔ (∀ (a : M), z * a = a * z) ∧ (∀ (b c : M), z * (b * c) = z * b * c) ∧ (∀ (a c : M), a * z * c = a * (z * c)) ∧ ∀ (a b : M), a * b * z = a * (b * z)

theorem IsAddCentral.left_comm {M : Type u_1} {a : M} [Add M] (h : IsAddCentral a) (b : M) (c : M) : a + (b + c) = b + (a + c)

theorem IsMulCentral.left_comm {M : Type u_1} {a : M} [Mul M] (h : IsMulCentral a) (b : M) (c : M) : a * (b * c) = b * (a * c)

theorem IsAddCentral.right_comm {M : Type u_1} {c : M} [Add M] (h : IsAddCentral c) (a : M) (b : M) : a + b + c = a + c + b

theorem IsMulCentral.right_comm {M : Type u_1} {c : M} [Mul M] (h : IsMulCentral c) (a : M) (b : M) : a * b * c = a * c * b

def Set.addCenter (M : Type u_1) [Add M] : Set M

The center of an additive magma.

Equations

• Set.addCenter M = {z : M | IsAddCentral z}

def Set.center (M : Type u_1) [Mul M] : Set M

The center of a magma.

Equations

• Set.center M = {z : M | IsMulCentral z}

theorem Set.mem_addCenter_iff (M : Type u_1) [Add M] {z : M} : z ∈ Set.addCenter M ↔ IsAddCentral z

theorem Set.mem_center_iff (M : Type u_1) [Mul M] {z : M} : z ∈ Set.center M ↔ IsMulCentral z

@[simp]

theorem Set.add_mem_addCenter {M : Type u_1} [Add M] {z₁ : M} {z₂ : M} (hz₁ : z₁ ∈ Set.addCenter M) (hz₂ : z₂ ∈ Set.addCenter M) : z₁ + z₂ ∈ Set.addCenter M

@[simp]

theorem Set.mul_mem_center {M : Type u_1} [Mul M] {z₁ : M} {z₂ : M} (hz₁ : z₁ ∈ Set.center M) (hz₂ : z₂ ∈ Set.center M) : z₁ * z₂ ∈ Set.center M

theorem AddSemigroup.mem_center_iff {M : Type u_1} [AddSemigroup M] {z : M} : z ∈ Set.addCenter M ↔ ∀ (g : M), g + z = z + g

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

instance Set.decidableMemCenter (M : Type u_1) [Semigroup M] [(a : M) → Decidable (∀ (b : M), b * a = a * b)] : DecidablePred fun (x : M) => x ∈ Set.center M

Equations

• Set.decidableMemCenter M x = decidable_of_iff' (∀ (g : M), g * x = x * g) ⋯

@[simp]

theorem Set.addCenter_eq_univ (M : Type u_1) [AddCommSemigroup M] : Set.addCenter M = Set.univ

@[simp]

theorem Set.center_eq_univ (M : Type u_1) [CommSemigroup M] : Set.center M = Set.univ

@[simp]

theorem Set.zero_mem_addCenter (M : Type u_1) [AddZeroClass M] : 0 ∈ Set.addCenter M

@[simp]

theorem Set.one_mem_center (M : Type u_1) [MulOneClass M] : 1 ∈ Set.center M

@[simp]

theorem Set.zero_mem_center (M : Type u_1) [MulZeroClass M] : 0 ∈ Set.center M

@[simp]

theorem Set.neg_mem_addCenter {M : Type u_1} [AddGroup M] {a : M} (ha : a ∈ Set.addCenter M) : -a ∈ Set.addCenter M

@[simp]

theorem Set.inv_mem_center {M : Type u_1} [Group M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M

theorem Set.subset_addCenter_add_units {M : Type u_1} [AddMonoid M] : AddUnits.val ⁻¹' Set.addCenter M ⊆ Set.addCenter (AddUnits M)

theorem Set.subset_center_units {M : Type u_1} [Monoid M] : Units.val ⁻¹' Set.center M ⊆ Set.center Mˣ

theorem Set.center_units_subset {M : Type u_1} [GroupWithZero M] : Set.center Mˣ ⊆ Units.val ⁻¹' Set.center M

theorem Set.center_units_eq {M : Type u_1} [GroupWithZero M] : Set.center Mˣ = Units.val ⁻¹' Set.center M

In a group with zero, the center of the units is the preimage of the center.

@[simp]

theorem Set.units_inv_mem_center {M : Type u_1} [Monoid M] {a : Mˣ} (ha : ↑a ∈ Set.center M) : ↑a⁻¹ ∈ Set.center M

@[simp]

theorem Set.invOf_mem_center {M : Type u_1} [Monoid M] {a : M} [Invertible a] (ha : a ∈ Set.center M) : ⅟a ∈ Set.center M

@[simp]

theorem Set.inv_mem_center₀ {M : Type u_1} [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M

@[simp]

theorem Set.sub_mem_addCenter {M : Type u_1} [AddGroup M] {a : M} {b : M} (ha : a ∈ Set.addCenter M) (hb : b ∈ Set.addCenter M) : a - b ∈ Set.addCenter M

@[simp]

theorem Set.div_mem_center {M : Type u_1} [Group M] {a : M} {b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) : a / b ∈ Set.center M

@[simp]

theorem Set.div_mem_center₀ {M : Type u_1} [GroupWithZero M] {a : M} {b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) : a / b ∈ Set.center M