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.

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

The center of an additive magma.

Equations

• Set.addCenter M = {z | ∀ (m : M), m + z = z + m}

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

The center of a magma.

Equations

• Set.center M = {z | ∀ (m : M), m * z = z * m}

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

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

instance Set.decidableMemCenter (M : Type u_1) [Mul M] [(a : M) → Decidable (∀ (b : M), b * a = a * b)] : DecidablePred fun x => 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.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.add_mem_addCenter {M : Type u_1} [AddSemigroup M] {a : M} {b : M} (ha : a ∈ Set.addCenter M) (hb : b ∈ Set.addCenter M) : a + b ∈ Set.addCenter M

@[simp]

theorem Set.mul_mem_center {M : Type u_1} [Semigroup 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_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

@[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

def AddSubsemigroup.center (M : Type u_1) [AddSemigroup 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) [Semigroup 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) }

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 ∈ AddSubsemigroup.center 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 ∈ Subsemigroup.center M ↔ ∀ (g : M), g * a = a * g)

theorem AddSubsemigroup.center.addCommSemigroup.proof_3 {M : Type u_1} [AddSemigroup M] : ∀ (x b : { x // x ∈ AddSubsemigroup.center M }), x + b = b + x

theorem AddSubsemigroup.center.addCommSemigroup.proof_1 {M : Type u_1} [AddSemigroup M] : AddMemClass (AddSubsemigroup M) M

theorem AddSubsemigroup.center.addCommSemigroup.proof_2 {M : Type u_1} [AddSemigroup M] (a : { x // x ∈ AddSubsemigroup.center M }) (b : { x // x ∈ AddSubsemigroup.center M }) (c : { x // x ∈ AddSubsemigroup.center M }) : a + b + c = a + (b + c)

instance AddSubsemigroup.center.addCommSemigroup {M : Type u_1} [AddSemigroup M] : AddCommSemigroup { x // x ∈ AddSubsemigroup.center M }

The center of an additive semigroup is commutative.

Equations

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

instance Subsemigroup.center.commSemigroup {M : Type u_1} [Semigroup M] : CommSemigroup { x // x ∈ Subsemigroup.center M }

The center of a semigroup is commutative.

Equations

• Subsemigroup.center.commSemigroup = let src := MulMemClass.toSemigroup (Subsemigroup.center M); CommSemigroup.mk (_ : ∀ (x b : { x // x ∈ Subsemigroup.center M }), x * b = b * x)

@[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 = ⊤