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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.

The centralizer of a subset of a semigroup M.

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      The centralizer of a subset of an additive semigroup.

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          @[simp]
          theorem Subsemigroup.coe_centralizer {M : Type u_1} (S : Set M) [Semigroup M] :
          theorem Subsemigroup.mem_centralizer_iff {M : Type u_1} {S : Set M} [Semigroup M] {z : M} :
          z centralizer S gS, g * z = z * g
          theorem AddSubsemigroup.mem_centralizer_iff {M : Type u_1} {S : Set M} [AddSemigroup M] {z : M} :
          z centralizer S gS, g + z = z + g
          instance Subsemigroup.decidableMemCentralizer {M : Type u_1} {S : Set M} [Semigroup M] (a : M) [Decidable (∀ bS, b * a = a * b)] :
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            instance AddSubsemigroup.decidableMemCentralizer {M : Type u_1} {S : Set M} [AddSemigroup M] (a : M) [Decidable (∀ bS, b + a = a + b)] :
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              theorem Subsemigroup.centralizer_le {M : Type u_1} {S T : Set M} [Semigroup M] (h : S T) :
              @[reducible, inline]
              abbrev Subsemigroup.closureCommSemigroupOfComm (M : Type u_1) [Semigroup M] {s : Set M} (hcomm : as, bs, a * b = b * a) :

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

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                  @[reducible, inline]
                  abbrev AddSubsemigroup.closureAddCommSemigroupOfComm (M : Type u_1) [AddSemigroup M] {s : Set M} (hcomm : as, bs, a + b = b + a) :

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

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