# Why is every singleton subset of a partially ordered set a totally ordered set?

In Lipschutz, Theory and Problems of Set Theory ( Schaum's series, 1964 ed.), Chapter 10, Solved Problem n°7 , the question is asked to find all non-empty totally ordered subsets of an ordered set A = {a,b,c} with a diagram showing that : b precedes a and that c precedes a ( b and c being incomparable).

Lipschutz provides a solution in which he gives {a} , {b} and {c} as totally ordered subsets of A.

Hence my question : is every singleton subset of a partially ordered set a totally ordered set.

How to explain that , in case a subset is a singleton the trichotomy condition for totally ordered sets is automatically fullfilled : a partial order S is total iff for all a and b belonging to S,

                      either a<b or a=b or a>b.


I'd like to derive the conclusion formally from the definition of a totally ordered set.

• It's because $\forall a,b: a=b$ holds in a singleton. – Berci Apr 29 at 12:27
• $a \in \{ a \}$ is the only element and $a=a$. Thus, the trichotomy condition is fulfilled. Recall that the condition id $\forall a,b \in X \ [a=b \lor \ldots ]$. – Mauro ALLEGRANZA Apr 29 at 12:28

1. Let $$S$$ be a singleton.
2. From $$1$$, there follows there must be some $$x$$ such that $$S=\{x\}$$.
3. Let $$a, b\in S$$.
4. Therefore, $$a=x$$ and $$b=x$$.
5. Therefore, $$a=b$$.
6. Therefore, $$a=b\lor ab$$.
• Very frustrating if nice answers like this are downvoted. And this without any explanation why. – drhab Apr 30 at 8:34
• @dmab. Totally agree. I personnaly upvoted both aswers. – Eleonore Saint James Apr 30 at 9:25

Yes, every singleton can be recognized as a totally ordered set (even as a well-ordered set).

This because next to reflexivity, transitivity and antisymmetry we also have comparability.

If $$S=\{c\}$$ is a singleton then it is true that for all elements $$a,b\in S$$ we have: $$ab$$

Actually in all cases (there is only one) we have $$a=c=b$$.

Also note that every non-empty subset of $$S$$ (there is only one) has a smallest element.