Documentation

Mathlib.Algebra.Order.Monoid.Basic

Ordered monoids #

This file develops some additional material on ordered monoids.

@[reducible, inline]
abbrev Function.Injective.orderedAddCommMonoid {α : Type u} [OrderedAddCommMonoid α] {β : Type u_2} [Zero β] [Add β] [SMul β] (f : βα) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ), f (n x) = n f x) :

Pullback an OrderedAddCommMonoid under an injective map.

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@[reducible, inline]
abbrev Function.Injective.orderedCommMonoid {α : Type u} [OrderedCommMonoid α] {β : Type u_2} [One β] [Mul β] [Pow β ] (f : βα) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) :

Pullback an OrderedCommMonoid under an injective map. See note [reducible non-instances].

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@[reducible, inline]
abbrev Function.Injective.orderedCancelAddCommMonoid {α : Type u} {β : Type u_1} [OrderedCancelAddCommMonoid α] [Zero β] [Add β] [SMul β] (f : βα) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ), f (n x) = n f x) :

Pullback an OrderedCancelAddCommMonoid under an injective map.

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@[reducible, inline]
abbrev Function.Injective.orderedCancelCommMonoid {α : Type u} {β : Type u_1} [OrderedCancelCommMonoid α] [One β] [Mul β] [Pow β ] (f : βα) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) :

Pullback an OrderedCancelCommMonoid under an injective map. See note [reducible non-instances].

Equations
@[reducible, inline]
abbrev Function.Injective.linearOrderedAddCommMonoid {α : Type u} [LinearOrderedAddCommMonoid α] {β : Type u_2} [Zero β] [Add β] [SMul β] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ), f (n x) = n f x) (sup : ∀ (x y : β), f (x y) = max (f x) (f y)) (inf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

Pullback an OrderedAddCommMonoid under an injective map.

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@[reducible, inline]
abbrev Function.Injective.linearOrderedCommMonoid {α : Type u} [LinearOrderedCommMonoid α] {β : Type u_2} [One β] [Mul β] [Pow β ] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (sup : ∀ (x y : β), f (x y) = max (f x) (f y)) (inf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

Pullback a LinearOrderedCommMonoid under an injective map. See note [reducible non-instances].

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@[reducible, inline]
abbrev Function.Injective.linearOrderedCancelAddCommMonoid {α : Type u} {β : Type u_1} [LinearOrderedCancelAddCommMonoid α] [Zero β] [Add β] [SMul β] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ), f (n x) = n f x) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

Pullback a LinearOrderedCancelAddCommMonoid under an injective map.

Equations
  • One or more equations did not get rendered due to their size.
@[reducible, inline]
abbrev Function.Injective.linearOrderedCancelCommMonoid {α : Type u} {β : Type u_1} [LinearOrderedCancelCommMonoid α] [One β] [Mul β] [Pow β ] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

Pullback a LinearOrderedCancelCommMonoid under an injective map. See note [reducible non-instances].

Equations
  • One or more equations did not get rendered due to their size.
def OrderEmbedding.addLeft {α : Type u_2} [Add α] [LinearOrder α] [AddLeftStrictMono α] (m : α) :
α ↪o α

The order embedding sending b to a + b, for some fixed a. See also OrderIso.addLeft when working in an additive ordered group.

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@[simp]
theorem OrderEmbedding.addLeft_apply {α : Type u_2} [Add α] [LinearOrder α] [AddLeftStrictMono α] (m : α) (n : α) :
@[simp]
theorem OrderEmbedding.mulLeft_apply {α : Type u_2} [Mul α] [LinearOrder α] [MulLeftStrictMono α] (m : α) (n : α) :
def OrderEmbedding.mulLeft {α : Type u_2} [Mul α] [LinearOrder α] [MulLeftStrictMono α] (m : α) :
α ↪o α

The order embedding sending b to a * b, for some fixed a. See also OrderIso.mulLeft when working in an ordered group.

Equations
def OrderEmbedding.addRight {α : Type u_2} [Add α] [LinearOrder α] [AddRightStrictMono α] (m : α) :
α ↪o α

The order embedding sending b to b + a, for some fixed a. See also OrderIso.addRight when working in an additive ordered group.

Equations
@[simp]
theorem OrderEmbedding.addRight_apply {α : Type u_2} [Add α] [LinearOrder α] [AddRightStrictMono α] (m : α) (n : α) :
@[simp]
theorem OrderEmbedding.mulRight_apply {α : Type u_2} [Mul α] [LinearOrder α] [MulRightStrictMono α] (m : α) (n : α) :
def OrderEmbedding.mulRight {α : Type u_2} [Mul α] [LinearOrder α] [MulRightStrictMono α] (m : α) :
α ↪o α

The order embedding sending b to b * a, for some fixed a. See also OrderIso.mulRight when working in an ordered group.

Equations