Are the empty set and a singleton set totally ordered sets?
I think yes, but I'm struggling to name the unique linear order for them.
This is what I think, I understand.
To a set A be totally ordered or linear ordered, these two conditions need to hold.
- R is transitive
- R has the trichotomy property over A
For the empty set, since is empty, I can't say that it doesn't hold those conditions. {} x {} = {} So, I think the unique order for an empty set is an empty set, but how do I write an ordered pair?
For the case of a singleton. Since a singleton set only has one element, I don't have another number to compare it with, so the trichotomy property doesn't hold.
1! = 1, so it has a unique order, But how do I write it?
Please explain. Thank you.
Getting the details right here depends on whether you think of orders in terms of the strict order relation $x < y$ or the weak order relation $x \le y$. For both the empty set and the singleton the strict order is empty, in the sense that as a relation thought of as a subset of $X \times X$ it's the empty set. The empty relation is vacuously transitive and vacuously satisfies trichotomy. For the singleton the unique weak order relation is equality, and for the empty set even the weak order relation is still empty.
Note that if $X$ is empty then so is $X \times X$, so the empty set has a unique relation on it, and this relation satisfies a bunch of conditions vacuously. If $X$ is the singleton then so is $X \times X$, so the singleton has exactly two relations on it, one of which is empty and one of which is the equality relation.
You can think of every set as having three canonically defined relations on it: the empty relation, the equality relation, and the "complete" relation (I don't know if there's a standard term for this) corresponding to all of $X \times X$. It's a funny fact about the singleton that the equality and complete relations are the same, and it's a funny fact about the empty set that all three relations are the same. It's a third funny fact that none of these relations define strict or weak orders on a set with more than one element.