Documentation

Mathlib.Algebra.Order.Ring.Cone

Constructing an ordered ring from a ring with a specified positive cone. #

Positive cones #

structure Ring.PositiveCone (α : Type u_2) [Ring α] extends AddCommGroup.PositiveCone :
Type u_2

A positive cone in a ring consists of a positive cone in underlying AddCommGroup, which contains 1 and such that the positive elements are closed under multiplication.

Instances For
    structure Ring.TotalPositiveCone (α : Type u_2) [Ring α] extends Ring.PositiveCone :
    Type u_2

    A total positive cone in a nontrivial ring induces a linear order.

    Instances For
      @[reducible]

      Forget that a TotalPositiveCone in a ring respects the multiplicative structure.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For

        Construct a StrictOrderedRing by designating a positive cone in an existing Ring.

        Equations
        • One or more equations did not get rendered due to their size.
        Instances For

          Construct a LinearOrderedRing by designating a positive cone in an existing Ring.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For