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Mathlib.GroupTheory.Submonoid.Centralizer

Centralizers of magmas and monoids #

Main definitions #

We provide Subgroup.centralizer, AddSubgroup.centralizer in other files.

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def Submonoid.centralizer {M : Type u_1} (S : Set M) [Monoid M] :
Submonoid M

The centralizer of a subset of a monoid M.

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      def AddSubmonoid.centralizer {M : Type u_1} (S : Set M) [AddMonoid M] :
      AddSubmonoid M

      The centralizer of a subset of an additive monoid.

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          @[simp]
          theorem Submonoid.coe_centralizer {M : Type u_1} (S : Set M) [Monoid M] :
          ↑(centralizer S) = S.centralizer
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          @[simp]
          theorem AddSubmonoid.coe_centralizer {M : Type u_1} (S : Set M) [AddMonoid M] :
          ↑(centralizer S) = S.addCentralizer
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          theorem Submonoid.centralizer_toSubsemigroup {M : Type u_1} (S : Set M) [Monoid M] :
          (centralizer S).toSubsemigroup = Subsemigroup.centralizer S
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          theorem AddSubmonoid.centralizer_toAddSubsemigroup {M : Type u_1} (S : Set M) [AddMonoid M] :
          (centralizer S).toAddSubsemigroup = AddSubsemigroup.centralizer S
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          theorem Submonoid.mem_centralizer_iff {M : Type u_1} {S : Set M} [Monoid M] {z : M} :
          z ∈ centralizer S ↔ ∀ g ∈ S, g * z = z * g
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          theorem AddSubmonoid.mem_centralizer_iff {M : Type u_1} {S : Set M} [AddMonoid M] {z : M} :
          z ∈ centralizer S ↔ ∀ g ∈ S, g + z = z + g
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          theorem Submonoid.center_le_centralizer {M : Type u_1} [Monoid M] (s : Set M) :
          center M ≤ centralizer s
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          theorem AddSubmonoid.center_le_centralizer {M : Type u_1} [AddMonoid M] (s : Set M) :
          center M ≤ centralizer s
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          instance Submonoid.decidableMemCentralizer {M : Type u_1} {S : Set M} [Monoid M] (a : M) [Decidable (∀ b ∈ S, b * a = a * b)] :
          Decidable (a ∈ centralizer S)
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            instance AddSubmonoid.decidableMemCentralizer {M : Type u_1} {S : Set M} [AddMonoid M] (a : M) [Decidable (∀ b ∈ S, b + a = a + b)] :
            Decidable (a ∈ centralizer S)
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              theorem Submonoid.centralizer_le {M : Type u_1} {S T : Set M} [Monoid M] (h : S ⊆ T) :
              centralizer T ≤ centralizer S
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              theorem AddSubmonoid.centralizer_le {M : Type u_1} {S T : Set M} [AddMonoid M] (h : S ⊆ T) :
              centralizer T ≤ centralizer S
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              @[simp]
              theorem Submonoid.centralizer_eq_top_iff_subset {M : Type u_1} [Monoid M] {s : Set M} :
              centralizer s = ⊤ ↔ s ⊆ ↑(center M)
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              @[simp]
              theorem AddSubmonoid.centralizer_eq_top_iff_subset {M : Type u_1} [AddMonoid M] {s : Set M} :
              centralizer s = ⊤ ↔ s ⊆ ↑(center M)
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              @[simp]
              theorem Submonoid.centralizer_univ (M : Type u_1) [Monoid M] :
              centralizer Set.univ = center M
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              @[simp]
              theorem AddSubmonoid.centralizer_univ (M : Type u_1) [AddMonoid M] :
              centralizer Set.univ = center M
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              theorem Submonoid.le_centralizer_centralizer (M : Type u_1) [Monoid M] {s : Submonoid M} :
              s ≤ centralizer ↑(centralizer ↑s)
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              theorem AddSubmonoid.le_centralizer_centralizer (M : Type u_1) [AddMonoid M] {s : AddSubmonoid M} :
              s ≤ centralizer ↑(centralizer ↑s)
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              @[simp]
              theorem Submonoid.centralizer_centralizer_centralizer (M : Type u_1) [Monoid M] {s : Set M} :
              centralizer s.centralizer.centralizer = centralizer s
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              @[simp]
              theorem AddSubmonoid.centralizer_centralizer_centralizer (M : Type u_1) [AddMonoid M] {s : Set M} :
              centralizer s.addCentralizer.addCentralizer = centralizer s
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              theorem Submonoid.closure_le_centralizer_centralizer {M : Type u_1} [Monoid M] (s : Set M) :
              closure s ≤ centralizer ↑(centralizer s)
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              theorem AddSubmonoid.closure_le_centralizer_centralizer {M : Type u_1} [AddMonoid M] (s : Set M) :
              closure s ≤ centralizer ↑(centralizer s)
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              @[reducible, inline]
              abbrev Submonoid.closureCommMonoidOfComm (M : Type u_1) [Monoid M] {s : Set M} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
              CommMonoid ↥(closure s)

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

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                  @[reducible, inline]
                  abbrev AddSubmonoid.closureAddCommMonoidOfComm (M : Type u_1) [AddMonoid M] {s : Set M} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a + b = b + a) :
                  AddCommMonoid ↥(closure s)

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

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