Introduction
This post is just dumping a short list of notions of order from my MSc thesis days.
Broad Classes of Orders
There are four common classes of orders.
Non-Strict Partial Order
- Reflexive
- Antisymmetric
- Transitive
Strict Partial Order
- Irreflexive
- Asymmetric
- Transitive
Non-Strict Total Order
- Reflexive
- Transitive
- Antisymmetric
- Strongly-connected
Strict Total Order
- Irreflexive
- Transitive
- Connected
Multidimensional Orders
There are orders which sit over tuples, which themselves may be elements of a relation.
- Product order
- Lexicographic order
- Pareto order
Function-Induced Orders
Sometimes a collection can be assigned an order by the level sets of a function.
Consensus Order
Suppose a collection of points $S$ where $x \leq y$ if-and-only-if there are more elements of $x$ dominated by elements of $y$ than vice versa.
Proximity Order
We can define orders by their proximity to a reference element or set.
Point Proximity Orders
Give a choice of metric space $(X, \rho)$ and a reference element of that metric space $\psi \in X$, then we consider $x \leq y$ if-and-only-if $\rho(\psi, x) \leq \rho(\psi, y)$.
Set Proximity Orders
The Hausdorff distance can similarly be used to induce an order on a collection of sets given a reference set.