Semiconjugate elements of a semigroup

Main definitions

We say that x is semiconjugate to y by a (SemiconjBy a x y), if a * x = y * a. In this file we provide operations on SemiconjBy _ _ _.

In the names of these operations, we treat a as the “left” argument, and both x and y as “right” arguments. This way most names in this file agree with the names of the corresponding lemmas for Commute a b = SemiconjBy a b b. As a side effect, some lemmas have only _right version.

Lean does not immediately recognise these terms as equations, so for rewriting we need syntax like rw [(h.pow_right 5).eq] rather than just rw [h.pow_right 5].

This file provides only basic operations (mul_left, mul_right, inv_right etc). Other operations (pow_right, field inverse etc) are in the files that define corresponding notions.

def SemiconjBy {M : Type u_2} [Mul M] (a x y : M) : Prop

x is semiconjugate to y by a, if a * x = y * a.

Equations

• SemiconjBy a x y = (a * x = y * a)

def AddSemiconjBy {M : Type u_2} [Add M] (a x y : M) : Prop

x is additive semiconjugate to y by a if a + x = y + a

Equations

• AddSemiconjBy a x y = (a + x = y + a)

theorem SemiconjBy.eq {S : Type u_1} [Mul S] {a x y : S} (h : SemiconjBy a x y) : a * x = y * a

Equality behind SemiconjBy a x y; useful for rewriting.

theorem AddSemiconjBy.eq {S : Type u_1} [Add S] {a x y : S} (h : AddSemiconjBy a x y) : a + x = y + a

Equality behind AddSemiconjBy a x y; useful for rewriting.

@[simp]

theorem SemiconjBy.mul_right {S : Type u_1} [Semigroup S] {a x y x' y' : S} (h : SemiconjBy a x y) (h' : SemiconjBy a x' y') : SemiconjBy a (x * x') (y * y')

If a semiconjugates x to y and x' to y', then it semiconjugates x * x' to y * y'.

@[simp]

theorem AddSemiconjBy.add_right {S : Type u_1} [AddSemigroup S] {a x y x' y' : S} (h : AddSemiconjBy a x y) (h' : AddSemiconjBy a x' y') : AddSemiconjBy a (x + x') (y + y')

If a semiconjugates x to y and x' to y', then it semiconjugates x + x' to y + y'.

theorem SemiconjBy.mul_left {S : Type u_1} [Semigroup S] {a b x y z : S} (ha : SemiconjBy a y z) (hb : SemiconjBy b x y) : SemiconjBy (a * b) x z

If b semiconjugates x to y and a semiconjugates y to z, then a * b semiconjugates x to z.

theorem AddSemiconjBy.add_left {S : Type u_1} [AddSemigroup S] {a b x y z : S} (ha : AddSemiconjBy a y z) (hb : AddSemiconjBy b x y) : AddSemiconjBy (a + b) x z

If b semiconjugates x to y and a semiconjugates y to z, then a + b semiconjugates x to z.

theorem SemiconjBy.transitive {S : Type u_1} [Semigroup S] : Transitive fun (a b : S) => ∃ (c : S), SemiconjBy c a b

The relation “there exists an element that semiconjugates a to b” on a semigroup is transitive.

theorem AddSemiconjBy.transitive {S : Type u_1} [AddSemigroup S] : Transitive fun (a b : S) => ∃ (c : S), AddSemiconjBy c a b

The relation “there exists an element that semiconjugates a to b” on an additive semigroup is transitive.

@[simp]

theorem SemiconjBy.one_right {M : Type u_2} [MulOneClass M] (a : M) : SemiconjBy a 1 1

Any element semiconjugates 1 to 1.

@[simp]

theorem AddSemiconjBy.zero_right {M : Type u_2} [AddZeroClass M] (a : M) : AddSemiconjBy a 0 0

Any element semiconjugates 0 to 0.

@[simp]

theorem SemiconjBy.one_left {M : Type u_2} [MulOneClass M] (x : M) : SemiconjBy 1 x x

One semiconjugates any element to itself.

@[simp]

theorem AddSemiconjBy.zero_left {M : Type u_2} [AddZeroClass M] (x : M) : AddSemiconjBy 0 x x

Zero semiconjugates any element to itself.

theorem SemiconjBy.reflexive {M : Type u_2} [MulOneClass M] : Reflexive fun (a b : M) => ∃ (c : M), SemiconjBy c a b

The relation “there exists an element that semiconjugates a to b” on a monoid (or, more generally, on MulOneClass type) is reflexive.

theorem AddSemiconjBy.reflexive {M : Type u_2} [AddZeroClass M] : Reflexive fun (a b : M) => ∃ (c : M), AddSemiconjBy c a b

The relation “there exists an element that semiconjugates a to b” on an additive monoid (or, more generally, on an AddZeroClass type) is reflexive.

@[simp]

theorem SemiconjBy.pow_right {M : Type u_2} [Monoid M] {a x y : M} (h : SemiconjBy a x y) (n : ℕ) : SemiconjBy a (x ^ n) (y ^ n)

@[simp]

theorem AddSemiconjBy.nsmul_right {M : Type u_2} [AddMonoid M] {a x y : M} (h : AddSemiconjBy a x y) (n : ℕ) : AddSemiconjBy a (n • x) (n • y)

theorem SemiconjBy.conj_mk {G : Type u_3} [Group G] (a x : G) : SemiconjBy a x (a * x * a⁻¹)

a semiconjugates x to a * x * a⁻¹.

theorem AddSemiconjBy.conj_mk {G : Type u_3} [AddGroup G] (a x : G) : AddSemiconjBy a x (a + x + -a)

a semiconjugates x to a + x + -a.

@[simp]

theorem SemiconjBy.conj_iff {G : Type u_3} [Group G] {a x y b : G} : SemiconjBy (b * a * b⁻¹) (b * x * b⁻¹) (b * y * b⁻¹) ↔ SemiconjBy a x y

@[simp]

theorem AddSemiconjBy.conj_iff {G : Type u_3} [AddGroup G] {a x y b : G} : AddSemiconjBy (b + a + -b) (b + x + -b) (b + y + -b) ↔ AddSemiconjBy a x y

@[simp]

theorem semiconjBy_iff_eq {M : Type u_2} [CancelCommMonoid M] {a x y : M} : SemiconjBy a x y ↔ x = y

@[simp]

theorem addSemiconjBy_iff_eq {M : Type u_2} [AddCancelCommMonoid M] {a x y : M} : AddSemiconjBy a x y ↔ x = y