# mathlib3documentation

algebra.order.monoid.cancel.basic

# Basic results on ordered cancellative monoids. #

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We pull back ordered cancellative monoids along injective maps.

@[reducible]
def function.injective.ordered_cancel_add_comm_monoid {α : Type u} {β : Type u_1} [has_zero β] [has_add β] [ β] (f : β α) (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 ordered_cancel_add_comm_monoid under an injective map.

Equations
@[reducible]
def function.injective.ordered_cancel_comm_monoid {α : Type u} {β : Type u_1} [has_one β] [has_mul β] [ ] (f : β α) (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 ordered_cancel_comm_monoid under an injective map. See note [reducible non-instances].

Equations
@[reducible]
def function.injective.linear_ordered_cancel_comm_monoid {α : Type u} {β : Type u_1} [has_one β] [has_mul β] [ ] [has_sup β] [has_inf β] (f : β α) (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) = linear_order.max (f x) (f y)) (hinf : (x y : β), f (x y) = linear_order.min (f x) (f y)) :

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

Equations
@[reducible]
def function.injective.linear_ordered_cancel_add_comm_monoid {α : Type u} {β : Type u_1} [has_zero β] [has_add β] [ β] [has_sup β] [has_inf β] (f : β α) (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) = linear_order.max (f x) (f y)) (hinf : (x y : β), f (x y) = linear_order.min (f x) (f y)) :

Pullback a linear_ordered_cancel_add_comm_monoid under an injective map.

Equations