About Ordering Theory

Properties of various ordering

1. Preorder
   • reflexive
   • transitive
2. Partial order(non-strict)
   • reflexive
   • antisymmetric
   • transitive.
3. Partial order(strict)
   • irreflexive
   • antisymmetric
   • transitive.
4. Total order
   • total
   • antisymmetric
   • transivtive
5. Strict weak orderings(strict partial order and transitivityOfIncomparability)
   • irreflexive
   • antisymmetric
   • transitivity
   • transitivityOfIncomparability
6. Well-order(total order and every non-empty subset of S has a leat element)
   • total
   • antisymmetric
   • transivtive
   • leastOfSubset

Some definition

Suppose P is a set and R(<=) is a relation on P, then we have following definition, for all a, b, c in P:

1. reflexive: a<=a
2. antisymmetric: if a<=b and b<=a then a=b(if a<=b and a!=b then b<=a must NOT hold)
3. transitivity: if a<=b and b<=c then a<=c
4. totality: a<=b or b<=a (it deduces reflexive)
5. comparability(a, b): a<b or a<b
6. incomparability(a, b): neither a<b nor b<a
7. transitivityOfIncomparability: for all a, b ,c, if incomparability(a,b) and incomparability(b,c), then incomparability(a,c)
8. leastOfSubset: for all non-empty subset A of S, there exits a least element is A
9. irreflexive: not a<=a

About strict <==> non-strict

If $<=$ a non-strict partial order, then the corresponding non-strict partial order $<=$ is the reflexive closure given by: $a<b$ if $a<=b \wedge a!=b$.

Conversely, strict ==> non-strict: $a<=b$ if $a<b \vee a=b$.

Reference

Order theory

List of order theory topics