Order
An Order is a concept from Category Theory which is about ordering objects. A less strict version of an order is a Preorder. 1
It is composed of two things: 1
- An underlying set
- A binary relation
They follow the following laws: 1
- Reflexivity - Each object in the underlying set is either greater than or equal to itself
- Transitivity - if a >= b and b >= c, then a >= c
- Antisymmetry - The way you define the ordering cannot give contradictory results, such that there are no ties and that if x < y, then it is not true that y < x
If an order exhibits a special case of reflexivity known as totality, then the order is a total order. Otherwise, it is a partial order. [^1
Orders can have the following properties: 1
- Greatest object - The object that is definitively greater than all the others in a partial or total order.
- Smallest object - The object that is definitively smaller than all the others in a partial or total order.]