The Definition of an Ordered Field in Real Analysis

Ordered Field in Real Analysis

An ordered field is a mathematical structure that combines the properties of a field with an order relation that is compatible with the field operations. Here's a precise definition:

Definition:

An ordered field (F, +, ⋅, ≤) is a set F equipped with two binary operations, addition + and multiplication ⋅, and a total order ≤ such that:

  1. Field Properties:
    • Associativity: (a + b) + c = a + (b + c) and (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) for all a, b, c ∈ F.
    • Commutativity: a + b = b + a and a ⋅ b = b ⋅ a for all a, b ∈ F.
    • Distributivity: a ⋅ (b + c) = a ⋅ b + a ⋅ c for all a, b, c ∈ F.
    • Existence of additive identity: There exists an element 0 ∈ F such that a + 0 = a for all a ∈ F.
    • Existence of multiplicative identity: There exists an element 1 ∈ F (with 1 ≠ 0) such that a ⋅ 1 = a for all a ∈ F.
    • Additive inverses: For each a ∈ F, there exists an element -a ∈ F such that a + (-a) = 0.
    • Multiplicative inverses: For each a ∈ F \ {0}, there exists an element a-1 ∈ F such that a ⋅ a-1 = 1.
  2. Total Order Properties:
    • The relation ≤ is a total order on F. This means:
      • Reflexivity: a ≤ a for all a ∈ F.
      • Antisymmetry: If a ≤ b and b ≤ a, then a = b.
      • Transitivity: If a ≤ b and b ≤ c, then a ≤ c.
      • Totality: For all a, b ∈ F, either a ≤ b or b ≤ a.
  3. Compatibility with Field Operations:
    • Order preservation under addition: If a ≤ b, then a + c ≤ b + c for all c ∈ F.
    • Order preservation under multiplication by positives: If 0 ≤ a and 0 ≤ b, then 0 ≤ a ⋅ b.

Example:

The set of real numbers ℝ with the usual addition, multiplication, and the standard order ≤ is an ordered field. However, the set of complex numbers ℂ cannot be an ordered field because there is no way to define a total order on ℂ that satisfies the required compatibility with field operations.

Ordered fields are fundamental in real analysis because they provide the structure needed to define concepts like limits, continuity, and integration, all of which rely on the ability to compare the sizes of numbers and operate on them.

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