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:
- Reflexivity - Each object in the underlying set is either greater than or equal to itself. 1
- Transitivity - if a ≥ b and b ≥ c, then a ≥ c. 1
- 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. In other words, there cannot be any cycles. 2
If every pair of elements in an order can be compared, then it has the property known as totality and the order is considered a total order. Otherwise, it is a partial order. 1 2
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.]