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/- | ||
Copyright (c) 2018 Chris Hughes. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Chris Hughes, Yury Kudryashov | ||
-/ | ||
import Mathlib.Algebra.Group.Action.End | ||
import Mathlib.Algebra.Group.Equiv.Basic | ||
import Mathlib.Algebra.GroupWithZero.Action.Defs | ||
import Mathlib.Algebra.GroupWithZero.Action.Units | ||
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/-! | ||
# Group actions and (endo)morphisms | ||
-/ | ||
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assert_not_exists Equiv.Perm.equivUnitsEnd | ||
assert_not_exists Prod.fst_mul | ||
assert_not_exists Ring | ||
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open Function | ||
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variable {M N A B α β : Type*} | ||
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/-- Push forward the action of `R` on `M` along a compatible surjective map `f : R →* S`. | ||
See also `Function.Surjective.mulActionLeft` and `Function.Surjective.moduleLeft`. | ||
-/ | ||
abbrev Function.Surjective.distribMulActionLeft {R S M : Type*} [Monoid R] [AddMonoid M] | ||
[DistribMulAction R M] [Monoid S] [SMul S M] (f : R →* S) (hf : Function.Surjective f) | ||
(hsmul : ∀ (c) (x : M), f c • x = c • x) : DistribMulAction S M := | ||
{ hf.distribSMulLeft f hsmul, hf.mulActionLeft f hsmul with } | ||
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section AddMonoid | ||
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variable (A) [AddMonoid A] [Monoid M] [DistribMulAction M A] | ||
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/-- Compose a `DistribMulAction` with a `MonoidHom`, with action `f r' • m`. | ||
See note [reducible non-instances]. -/ | ||
abbrev DistribMulAction.compHom [Monoid N] (f : N →* M) : DistribMulAction N A := | ||
{ DistribSMul.compFun A f, MulAction.compHom A f with } | ||
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end AddMonoid | ||
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section Monoid | ||
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variable (A) [Monoid A] [Monoid M] [MulDistribMulAction M A] | ||
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/-- Compose a `MulDistribMulAction` with a `MonoidHom`, with action `f r' • m`. | ||
See note [reducible non-instances]. -/ | ||
abbrev MulDistribMulAction.compHom [Monoid N] (f : N →* M) : MulDistribMulAction N A := | ||
{ MulAction.compHom A f with | ||
smul_one := fun x => smul_one (f x), | ||
smul_mul := fun x => smul_mul' (f x) } | ||
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end Monoid | ||
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/-- The tautological action by `AddMonoid.End α` on `α`. | ||
This generalizes `Function.End.applyMulAction`. -/ | ||
instance AddMonoid.End.applyDistribMulAction [AddMonoid α] : | ||
DistribMulAction (AddMonoid.End α) α where | ||
smul := (· <| ·) | ||
smul_zero := AddMonoidHom.map_zero | ||
smul_add := AddMonoidHom.map_add | ||
one_smul _ := rfl | ||
mul_smul _ _ _ := rfl | ||
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@[simp] | ||
theorem AddMonoid.End.smul_def [AddMonoid α] (f : AddMonoid.End α) (a : α) : f • a = f a := | ||
rfl | ||
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/-- `AddMonoid.End.applyDistribMulAction` is faithful. -/ | ||
instance AddMonoid.End.applyFaithfulSMul [AddMonoid α] : | ||
FaithfulSMul (AddMonoid.End α) α := | ||
⟨fun {_ _ h} => AddMonoidHom.ext h⟩ | ||
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/-- Each non-zero element of a `GroupWithZero` defines an additive monoid isomorphism of an | ||
`AddMonoid` on which it acts distributively. | ||
This is a stronger version of `DistribMulAction.toAddMonoidHom`. -/ | ||
def DistribMulAction.toAddEquiv₀ {α : Type*} (β : Type*) [GroupWithZero α] [AddMonoid β] | ||
[DistribMulAction α β] (x : α) (hx : x ≠ 0) : β ≃+ β := | ||
{ DistribMulAction.toAddMonoidHom β x with | ||
invFun := fun b ↦ x⁻¹ • b | ||
left_inv := fun b ↦ inv_smul_smul₀ hx b | ||
right_inv := fun b ↦ smul_inv_smul₀ hx b } | ||
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variable (M A) in | ||
/-- Each element of the monoid defines a monoid homomorphism. -/ | ||
@[simps] | ||
def MulDistribMulAction.toMonoidEnd [Monoid M] [Monoid A] [MulDistribMulAction M A] : | ||
M →* Monoid.End A where | ||
toFun := MulDistribMulAction.toMonoidHom A | ||
map_one' := MonoidHom.ext <| one_smul M | ||
map_mul' x y := MonoidHom.ext <| mul_smul x y |
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/- | ||
Copyright (c) 2018 Chris Hughes. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Chris Hughes, Yury Kudryashov | ||
-/ | ||
import Mathlib.Algebra.Group.Action.Faithful | ||
import Mathlib.Algebra.GroupWithZero.NeZero | ||
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/-! | ||
# Faithful actions involving groups with zero | ||
-/ | ||
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assert_not_exists Equiv.Perm.equivUnitsEnd | ||
assert_not_exists Prod.fst_mul | ||
assert_not_exists Ring | ||
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open Function | ||
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variable {M N A B α β : Type*} | ||
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/-- `Monoid.toMulAction` is faithful on nontrivial cancellative monoids with zero. -/ | ||
instance CancelMonoidWithZero.faithfulSMul [CancelMonoidWithZero α] [Nontrivial α] : | ||
FaithfulSMul α α where eq_of_smul_eq_smul h := mul_left_injective₀ one_ne_zero (h 1) |
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