Is a complete flag well-ordered?

Question

Given a vector space $V$, a collection $\mathcal{F}$ of subspaces of $V$ is called a flag of $V$ if

• $\{0\}\in\mathcal{F}$
• $V\in\mathcal{F}$
• $\mathcal{F}$ is a chain

Furthermore, a flag $\mathcal{F}$ of $V$ is called a complete flag if $\mathcal{F}$ is not properly contained in any other flag of $V$, i.e. it is a maximal flag.

My question is:

Is every complete flag of a vector space well-ordered by $\subseteq$?

It is clearly true for finite dimensional vector spaces. I have been searching for a counter-example but have been drawing a blank. Any help is appreciated.

To show that a complete flag $\mathcal{F}$ is well-ordered, I need to show that for any $\mathcal{C}\subseteq\mathcal{F}$ that $\mathcal{C}$ has a least element. The natural candidate to test would be $\bigcap\mathcal{C}$. We know that $\bigcap\mathcal{C}$ is in $\mathcal{F}$, for if it weren't that would contradict $\mathcal{F}$'s maximality. Also, it is clearly the largest element of $\mathcal{F}$ that is still smaller than all elements of $\mathcal{C}$. Thus if I can find a collection of subspaces $\mathcal{C}$ contained in the maximal flag such that $\bigcap\mathcal{C}\not\in\mathcal{C}$ I will have found a counter-example.

Answer 1

Let $V$ be the space of all polynomials in one variable $x$ over a field $F$. Let $V_n$ be the subspace of $V$ consisting of all polynomials divisible by $x^n$. Then $V_{n+1}\subsetneq V_n$ for each $n$, so a maximal chain containing all the $V_n$ is a counterexample.

Answer 2

No.

First of all note that every flag can be extended to a complete flag. This is just a maximal chain of vector spaces.

Secondly, note that if $B$ is a basis, then any chain in $\mathcal P(B)$ will correspond a flag (complete or not).

Finally, note that if $B$ is infinite then in $\mathcal P(B)$ there is a chain which embeds $\Bbb Q$ (or more, much more). Therefore cannot be extended to a well-ordered chain. In particular the flag it induces cannot be extended to a complete flag which is well-ordered.