Centralizers in semigroups, as subsemigroups.

Main definitions

• Subsemigroup.centralizer: the centralizer of a subset of a semigroup
• AddSubsemigroup.centralizer: the centralizer of a subset of an additive semigroup

We provide Monoid.centralizer, AddMonoid.centralizer, Subgroup.centralizer, and AddSubgroup.centralizer in other files.

def Subsemigroup.centralizer {M : Type u_1} (S : Set M) [Semigroup M] : Subsemigroup M

The centralizer of a subset of a semigroup M.

Equations

• Subsemigroup.centralizer S = { carrier := S.centralizer, mul_mem' := ⋯ }

def AddSubsemigroup.centralizer {M : Type u_1} (S : Set M) [AddSemigroup M] : AddSubsemigroup M

The centralizer of a subset of an additive semigroup.

Equations

• AddSubsemigroup.centralizer S = { carrier := S.addCentralizer, add_mem' := ⋯ }

@[simp]

theorem Subsemigroup.coe_centralizer {M : Type u_1} (S : Set M) [Semigroup M] : ↑(centralizer S) = S.centralizer

@[simp]

theorem AddSubsemigroup.coe_centralizer {M : Type u_1} (S : Set M) [AddSemigroup M] : ↑(centralizer S) = S.addCentralizer

theorem Subsemigroup.mem_centralizer_iff {M : Type u_1} {S : Set M} [Semigroup M] {z : M} : z ∈ centralizer S ↔ ∀ g ∈ S, g * z = z * g

theorem AddSubsemigroup.mem_centralizer_iff {M : Type u_1} {S : Set M} [AddSemigroup M] {z : M} : z ∈ centralizer S ↔ ∀ g ∈ S, g + z = z + g

@[implicit_reducible]

instance Subsemigroup.decidableMemCentralizer {M : Type u_1} {S : Set M} [Semigroup M] (a : M) [Decidable (∀ b ∈ S, b * a = a * b)] : Decidable (a ∈ centralizer S)

Equations

• Subsemigroup.decidableMemCentralizer a = decidable_of_iff' (∀ g ∈ S, g * a = a * g) ⋯

@[implicit_reducible]

instance AddSubsemigroup.decidableMemCentralizer {M : Type u_1} {S : Set M} [AddSemigroup M] (a : M) [Decidable (∀ b ∈ S, b + a = a + b)] : Decidable (a ∈ centralizer S)

Equations

• AddSubsemigroup.decidableMemCentralizer a = decidable_of_iff' (∀ g ∈ S, g + a = a + g) ⋯

theorem Subsemigroup.center_le_centralizer {M : Type u_1} [Semigroup M] (S : Set M) : center M ≤ centralizer S

theorem AddSubsemigroup.center_le_centralizer {M : Type u_1} [AddSemigroup M] (S : Set M) : center M ≤ centralizer S

theorem Subsemigroup.centralizer_le {M : Type u_1} {S T : Set M} [Semigroup M] (h : S ⊆ T) : centralizer T ≤ centralizer S

theorem AddSubsemigroup.centralizer_le {M : Type u_1} {S T : Set M} [AddSemigroup M] (h : S ⊆ T) : centralizer T ≤ centralizer S

@[simp]

theorem Subsemigroup.centralizer_eq_top_iff_subset {M : Type u_1} [Semigroup M] {s : Set M} : centralizer s = ⊤ ↔ s ⊆ ↑(center M)

@[simp]

theorem AddSubsemigroup.centralizer_eq_top_iff_subset {M : Type u_1} [AddSemigroup M] {s : Set M} : centralizer s = ⊤ ↔ s ⊆ ↑(center M)

@[simp]

theorem Subsemigroup.centralizer_univ (M : Type u_1) [Semigroup M] : centralizer Set.univ = center M

@[simp]

theorem AddSubsemigroup.centralizer_univ (M : Type u_1) [AddSemigroup M] : centralizer Set.univ = center M

theorem Subsemigroup.closure_le_centralizer_centralizer {M : Type u_1} [Semigroup M] (s : Set M) : closure s ≤ centralizer ↑(centralizer s)

theorem AddSubsemigroup.closure_le_centralizer_centralizer {M : Type u_1} [AddSemigroup M] (s : Set M) : closure s ≤ centralizer ↑(centralizer s)

theorem Subsemigroup.isMulCommutative_closure (M : Type u_1) [Semigroup M] {s : Set M} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) : IsMulCommutative ↥(closure s)

If all the elements of a set s commute, then closure s is commutative.

theorem AddSubsemigroup.isAddCommutative_closure (M : Type u_1) [AddSemigroup M] {s : Set M} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a + b = b + a) : IsAddCommutative ↥(closure s)

If all the elements of a set s commute, then closure s is commutative.

@[reducible, inline, deprecated Subsemigroup.isMulCommutative_closure (since := "2026-03-09")]

abbrev Subsemigroup.closureCommSemigroupOfComm (M : Type u_1) [Semigroup M] {s : Set M} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) : CommSemigroup ↥(closure s)

If all the elements of a set s commute, then closure s is a commutative semigroup.

Equations

• Subsemigroup.closureCommSemigroupOfComm M hcomm = inferInstance

@[reducible, inline, deprecated Subsemigroup.isMulCommutative_closure (since := "2026-03-09")]

abbrev AddSubsemigroup.closureAddCommSemigroupOfComm (M : Type u_1) [AddSemigroup M] {s : Set M} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a + b = b + a) : AddCommSemigroup ↥(closure s)

If all the elements of a set s commute, then closure s forms an additive commutative semigroup.

Equations

• AddSubsemigroup.closureAddCommSemigroupOfComm M hcomm = inferInstance

instance Subsemigroup.instIsMulCommutative_closure (M : Type u_1) [Semigroup M] {S : Type u_2} [SetLike S M] [MulMemClass S M] (s : S) [IsMulCommutative ↥s] : IsMulCommutative ↥(closure ↑s)

instance AddSubsemigroup.instIsAddCommutative_closure (M : Type u_1) [AddSemigroup M] {S : Type u_2} [SetLike S M] [AddMemClass S M] (s : S) [IsAddCommutative ↥s] : IsAddCommutative ↥(closure ↑s)