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Mathlib.GroupTheory.Subsemigroup.Centralizer

Centralizers in semigroups, as subsemigroups. #

Main definitions #

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

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

The centralizer of a subset of a semigroup M.

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

    The centralizer of a subset of an additive semigroup.

    Instances For
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      @[simp]
      theorem Subsemigroup.coe_centralizer {M : Type u_1} (S : Set M) [Semigroup M] :
      ↑(centralizer S) = S.centralizer
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      @[simp]
      theorem AddSubsemigroup.coe_centralizer {M : Type u_1} (S : Set M) [AddSemigroup M] :
      ↑(centralizer S) = S.addCentralizer
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      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
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      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
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      @[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)
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      @[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)
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      theorem Subsemigroup.center_le_centralizer {M : Type u_1} [Semigroup M] (S : Set M) :
      center M ≤ centralizer S
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      theorem AddSubsemigroup.center_le_centralizer {M : Type u_1} [AddSemigroup M] (S : Set M) :
      center M ≤ centralizer S
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      theorem Subsemigroup.centralizer_le {M : Type u_1} {S T : Set M} [Semigroup M] (h : S ⊆ T) :
      centralizer T ≤ centralizer S
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      theorem AddSubsemigroup.centralizer_le {M : Type u_1} {S T : Set M} [AddSemigroup M] (h : S ⊆ T) :
      centralizer T ≤ centralizer S
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      @[simp]
      theorem Subsemigroup.centralizer_eq_top_iff_subset {M : Type u_1} [Semigroup M] {s : Set M} :
      centralizer s = ⊤ ↔ s ⊆ ↑(center M)
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      @[simp]
      theorem AddSubsemigroup.centralizer_eq_top_iff_subset {M : Type u_1} [AddSemigroup M] {s : Set M} :
      centralizer s = ⊤ ↔ s ⊆ ↑(center M)
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      @[simp]
      theorem Subsemigroup.centralizer_univ (M : Type u_1) [Semigroup M] :
      centralizer Set.univ = center M
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      @[simp]
      theorem AddSubsemigroup.centralizer_univ (M : Type u_1) [AddSemigroup M] :
      centralizer Set.univ = center M
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      theorem Subsemigroup.closure_le_centralizer_centralizer {M : Type u_1} [Semigroup M] (s : Set M) :
      closure s ≤ centralizer ↑(centralizer s)
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      theorem AddSubsemigroup.closure_le_centralizer_centralizer {M : Type u_1} [AddSemigroup M] (s : Set M) :
      closure s ≤ centralizer ↑(centralizer s)
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      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.

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      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.

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      @[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.

      Instances For
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        @[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.

        Instances For
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          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)
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          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)