WAT: opCmp and opEquals woes

[via Digitalmars-d]

In order to get an overview for myself I've summarized the 
various properties of different types of orders as they are 
described in Wikipedia (right or wrong). I post it here in case 
others have an interest in it (or corrections/extensions to it):

* Preorder:

a ≤ a (reflexivity)

if a ≤ b and b ≤ c then a ≤ c (transitivity)


* Non-strict Partial Order:

a ≤ a (reflexivity);

if a ≤ b and b ≤ c then a ≤ c (transitivity).

if a ≤ b and b ≤ a then a = b (antisymmetry);


* Strict Partial Order:

not a < a (irreflexivity),

if a < b and b < c then a < c (transitivity), and

if a < b then not b < a (asymmetry; implied by irreflexivity and 
transitivity).


* Total Order:

a ≤ b or b ≤ a (totality).

If a ≤ b and b ≤ c then a ≤ c (transitivity);

If a ≤ b and b ≤ a then a = b (antisymmetry);


* Pseudo-Order:

not (a < b and b < a) (antisymmetry)

if a < b then a < c or c < b (co-transivity/comparison)

if not (a < b or b < a) then a = b (equality)


a#b ===  a < b or b < a (apartness/negation of equality)
http://en.wikipedia.org/wiki/Apartness_relation


* Total Preorder:

x ≲ b or b ≲ a (totality).

if a ≲ b and b ≲ c then a ≲ c (transitivity).

x ≲ b (reflexivity; implied by transitivity and totality)


* Strict Weak Order: (complement of a total preorder)

not a < a (irreflexivity).

if a < b and b < c then a < c (transitivity).

if a < b then not b < a (asymmetry; implied by irreflexivity and 
transitivity).

if a is incomparable with y, and b is incomparable with z, then a 
is incomparable with c (transitivity of incomparability).