Centers of magmas and semigroups

Main definitions

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

References

• [Cabrera García and Rodríguez Palacios, Non-associative normed algebras. Volume 1] [cabreragarciarodriguezpalacios2014]

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 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_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 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) (_ : x ∈ Set.center M ↔ ∀ (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.natCast_mem_center (M : Type u_1) [NonAssocSemiring M] (n : ℕ) : ↑n ∈ Set.center M

@[simp]

theorem Set.ofNat_mem_center (M : Type u_1) [NonAssocSemiring M] (n : ℕ) [Nat.AtLeastTwo n] : OfNat.ofNat n ∈ Set.center M

@[simp]

theorem Set.intCast_mem_center (M : Type u_1) [NonAssocRing M] (n : ℤ) : ↑n ∈ 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

@[simp]

theorem Set.add_mem_center {M : Type u_1} [Distrib M] {a : M} {b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) : a + b ∈ Set.center M

@[simp]

theorem Set.neg_mem_center {M : Type u_1} [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) : -a ∈ 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

Set.center as a Subsemigroup.

def AddSubsemigroup.center (M : Type u_1) [Add M] : AddSubsemigroup M

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

Equations

• AddSubsemigroup.center M = { carrier := Set.addCenter M, add_mem' := (_ : ∀ {a b : M}, a ∈ Set.addCenter M → b ∈ Set.addCenter M → a + b ∈ Set.addCenter M) }

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

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

Equations

• Subsemigroup.center M = { carrier := Set.center M, mul_mem' := (_ : ∀ {a b : M}, a ∈ Set.center M → b ∈ Set.center M → a * b ∈ Set.center M) }

instance AddSubsemigroup.center.addCommSemigroup {M : Type u_1} [Add M] : AddCommSemigroup ↥(AddSubsemigroup.center M)

The center of an additive magma is commutative and associative.

Equations

• AddSubsemigroup.center.addCommSemigroup = AddCommSemigroup.mk (_ : ∀ (a x : ↥(AddSubsemigroup.center M)), a + x = x + a)

theorem AddSubsemigroup.center.addCommSemigroup.proof_2 {M : Type u_1} [Add M] (a : ↥(AddSubsemigroup.center M)) : ∀ (x : ↥(AddSubsemigroup.center M)), a + x = x + a

theorem AddSubsemigroup.center.addCommSemigroup.proof_1 {M : Type u_1} [Add M] : ∀ (x b x_1 : ↥(AddSubsemigroup.center M)), x + b + x_1 = x + (b + x_1)

instance Subsemigroup.center.commSemigroup {M : Type u_1} [Mul M] : CommSemigroup ↥(Subsemigroup.center M)

The center of a magma is commutative and associative.

Equations

• Subsemigroup.center.commSemigroup = CommSemigroup.mk (_ : ∀ (a x : ↥(Subsemigroup.center M)), a * x = x * a)

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

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

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

Equations

• AddSubsemigroup.decidableMemCenter a = decidable_of_iff' (∀ (g : M), g + a = a + g) (_ : a ∈ Set.addCenter M ↔ ∀ (g : M), g + a = a + g)

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

Equations

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

@[simp]

theorem AddSubsemigroup.center_eq_top (M : Type u_1) [AddCommSemigroup M] : AddSubsemigroup.center M = ⊤

@[simp]

theorem Subsemigroup.center_eq_top (M : Type u_1) [CommSemigroup M] : Subsemigroup.center M = ⊤