Can Partial Order have just one element?

Question

I came across the question that asks "Is it possible for a poset to have an element that is both maximum and minimum". So I can't imagine a element in poset being simultaneously be maximum and minimum, I guess poset must have element of 1.

If poset can have single elements then can Chain have one element as well? like if P = (X , <) is poset and X = {1,2,3,4,5} could I have chain of (1,1)?

Answer 1

Yes, there exists a partially ordered set (poset) with one element. In fact, there is only one such thing.

There even exists a poset with zero elements!

You are correct that any poset that has an element which is both a maximum and a minimum must be the one element poset. There is a simple formal proof:

Let $*$ denote the element that is both maximal and minimal. Then for any element $x$ of the poset:

• $x \leq *$, because $*$ is a maximum
• $* \leq x$, because $*$ is a minimum

and by the antisymmetry law, this implies $x = *$.