A poset can be represented by a directed graph where there is 0 or 1 (but not more than one) arrows between the nodes.
For example if we have a relation between two elements 'p' and 'q':
|p >= q||Then we write an arrow||p->q|
Non-strict partial order
- reflexivity: a ≤ a (every element is related to itself.)
- antisymmetry: if a ≤ b and b ≤ a then a = b
- transitivity: if a ≤ b and b ≤ c then a≤c.
A non-strict partial order is also known as an antisymmetric preorder.
Strict partial order
- Irreflexivity: not a < a (no element is related to itself)
- Transitivity: if a < b and b < c then a < c ,
- Asymmetry: if a < b then not b < a.
A strict partial order is also known as a strict preorder.
An isotone map preserves the ordering.
Fixpoint and W-completeness
W-complete - each well-ordered subset of P has a supremum.
A partially ordered set P is W-complete iff each selfmap of P has a fixpoint.
That is W-completeness implies a fixpoint and a fixpoint implies W-completeness.
To get an intuitive feel try to construct a map without a fixpoint.
So there must be at least one fixpoint.
Existence of minimal or maximal fixpoint