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 |

## Properties

See Wikipedia

#### 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.

## Isotone Maps

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. - The top element cant map to itself so let it go to the element below.
- The next element down cant map to the top element since an isotone map must preserve the ordering. So again it must map to (at least) the element below.
- If we continue like this then, sooner or later, we will run out of elements.
So there must be at least one fixpoint. |

#### Existence of minimal or maximal fixpoint