Semiconjugate and commuting maps

We define the following predicates:

function.semiconj: f : α → β semiconjugates ga : α → α to gb : β → β if f ∘ ga = gb ∘ f;
function.semiconj₂:f : α → βsemiconjugates a binary operationga : α → α → αtogb : β → β → βiff (ga x y) = gb (f x) (f y)`;
f : α → α commutes with g : α → α if f ∘ g = g ∘ f, or equivalently semiconj f g g.

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def function.semiconj {α : Type u_1} {β : Type u_2} (f : α → β) (ga : α → α) (gb : β → β) :

Prop

We say that f : α → β semiconjugates ga : α → α to gb : β → β if f ∘ ga = gb ∘ f.

Equations

function.semiconj f ga gb = ∀ (x : α), f (ga x) = gb (f x)

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theorem function.semiconj.comp_eq {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {gb : β → β} (h : function.semiconj f ga gb) :

f ∘ ga = gb ∘ f

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theorem function.semiconj.eq {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {gb : β → β} (h : function.semiconj f ga gb) (x : α) :

f (ga x) = gb (f x)

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theorem function.semiconj.comp_right {α : Type u_1} {β : Type u_2} {f : α → β} {ga ga' : α → α} {gb gb' : β → β} (h : function.semiconj f ga gb) (h' : function.semiconj f ga' gb') :

function.semiconj f (ga ∘ ga') (gb ∘ gb')

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theorem function.semiconj.comp_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} {fab : α → β} {fbc : β → γ} {ga : α → α} {gb : β → β} {gc : γ → γ} (hab : function.semiconj fab ga gb) (hbc : function.semiconj fbc gb gc) :

function.semiconj (fbc ∘ fab) ga gc

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theorem function.semiconj.id_right {α : Type u_1} {β : Type u_2} {f : α → β} :

function.semiconj f id id

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theorem function.semiconj.id_left {α : Type u_1} {ga : α → α} :

function.semiconj id ga ga

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theorem function.semiconj.inverses_right {α : Type u_1} {β : Type u_2} {f : α → β} {ga ga' : α → α} {gb gb' : β → β} (h : function.semiconj f ga gb) (ha : function.right_inverse ga' ga) (hb : function.left_inverse gb' gb) :

function.semiconj f ga' gb'

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def function.commute {α : Type u_1} (f g : α → α) :

Prop

Two maps f g : α → α commute if f ∘ g = g ∘ f.

Equations

function.commute f g = function.semiconj f g g

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theorem function.semiconj.commute {α : Type u_1} {f g : α → α} (h : function.semiconj f g g) :

function.commute f g

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theorem function.commute.refl {α : Type u_1} (f : α → α) :

function.commute f f

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theorem function.commute.symm {α : Type u_1} {f g : α → α} (h : function.commute f g) :

function.commute g f

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theorem function.commute.comp_right {α : Type u_1} {f g g' : α → α} (h : function.commute f g) (h' : function.commute f g') :

function.commute f (g ∘ g')

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theorem function.commute.comp_left {α : Type u_1} {f f' g : α → α} (h : function.commute f g) (h' : function.commute f' g) :

function.commute (f ∘ f') g

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theorem function.commute.id_right {α : Type u_1} {f : α → α} :

function.commute f id

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theorem function.commute.id_left {α : Type u_1} {f : α → α} :

function.commute id f

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def function.semiconj₂ {α : Type u_1} {β : Type u_2} (f : α → β) (ga : α → α → α) (gb : β → β → β) :

Prop

A map f semiconjugates a binary operation ga to a binary operation gb if for all x, y we have f (ga x y) = gb (f x) (f y). E.g., a monoid_hom semiconjugates (*) to (*).

Equations

function.semiconj₂ f ga gb = ∀ (x y : α), f (ga x y) = gb (f x) (f y)

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theorem function.semiconj₂.eq {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α → α} {gb : β → β → β} (h : function.semiconj₂ f ga gb) (x y : α) :

f (ga x y) = gb (f x) (f y)

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theorem function.semiconj₂.comp_eq {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α → α} {gb : β → β → β} (h : function.semiconj₂ f ga gb) :

function.bicompr f ga = function.bicompl gb f f

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theorem function.semiconj₂.id_left {α : Type u_1} (op : α → α → α) :

function.semiconj₂ id op op

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theorem function.semiconj₂.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {ga : α → α → α} {gb : β → β → β} {f' : β → γ} {gc : γ → γ → γ} (hf' : function.semiconj₂ f' gb gc) (hf : function.semiconj₂ f ga gb) :

function.semiconj₂ (f' ∘ f) ga gc