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A transitive, antisymmetric, and total (binary) relation on some set

Illustration

In programming, total order serves for comparison of values. For instance, in Language:Haskell, we may leverage total order on numbers as follows: 

> 41 < 42
True
> max 41 42
42

Some form of polymorphism may be used in many programming languages to define such a comparison-relate total order on given data types. For instance, in Haskell, there is a type class Ord for total order; its key member is compare which returns either LT, EQ, or GT. For instance: 

> compare 41 42
LT

Let us illustrate the definition a total order for a simple data type for natural numbers: 

-- Peano natural numbers
data Nat = Zero | Succ Nat

Before we define a total order for natural numbers, let us define equality, as it is effectively a precondition for total order. In Haskell, we instantiate the type class Eq hence: 

-- Equality of natural numbers
instance Eq Nat
  where
    Zero == Zero = True
    Zero == (Succ _) = False
    (Succ _) == Zero = False
    (Succ x) == (Succ y) = x == y

Thus, all pairs of constructor patterns are examined and accordingly mapped to truth values while subterms are processed recursively, when necessary. We can test for equality as follows: 

> Succ Zero == Zero
False
> Succ Zero == Succ Zero
True

The type-class instance for total order follows the same scheme: 

-- Total order on natural numbers
instance Ord Nat
  where
    compare Zero Zero = EQ
    compare Zero (Succ _) = LT
    compare (Succ _) Zero = GT
    compare (Succ x) (Succ y) = compare x y

We can test for total order as follows: 

> compare (Succ Zero) Zero
GT
> compare (Succ Zero) (Succ Zero)
EQ

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