Documentation

Mathlib.Algebra.Order.Monoid.Basic

Ordered monoids #

This file develops some additional material on ordered monoids.

@[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.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.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].

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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.linearOrderedCommMonoid {α : Type u} [LinearOrderedCommMonoid α] {β : Type u_2} [One β] [Mul β] [Pow β ] [Max β] [Min 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) = f x f y) (inf : ∀ (x y : β), f (x y) = 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.linearOrderedAddCommMonoid {α : Type u} [LinearOrderedAddCommMonoid α] {β : Type u_2} [Zero β] [Add β] [SMul β] [Max β] [Min 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) = f x f y) (inf : ∀ (x y : β), f (x y) = f x f y) :

            Pullback an OrderedAddCommMonoid under an injective map.

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              @[reducible, inline]
              abbrev Function.Injective.linearOrderedCancelCommMonoid {α : Type u} {β : Type u_1} [LinearOrderedCancelCommMonoid α] [One β] [Mul β] [Pow β ] [Max β] [Min 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) = f x f y) (hinf : ∀ (x y : β), f (x y) = f x f y) :

              Pullback a LinearOrderedCancelCommMonoid 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] [Max β] [Min 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) = f x f y) (hinf : ∀ (x y : β), f (x y) = f x f y) :

                Pullback a LinearOrderedCancelAddCommMonoid under an injective map.

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                  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.

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

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                        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.

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