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Let $(A,\leq)$ be a totally ordered abelian group, and $\Gamma\subseteq A$ be a set of nonnegative elements, such that it is wellordered by $\leq$. Is it true then that the semigroup $S$ generated by $\Gamma$ is again wellordered?

Thoughts so far:

  • Positivity hypothesis is obviously necessary, because for any negative $a\in A$, we have that $n\cdot a$ is an infinite decreasing sequence.
  • I can show that for a wellordered $\Gamma$, it is also true that $\Gamma+\Gamma$ is wellordered, which easily extends to sets $\Gamma_n:=\Gamma+\Gamma+\ldots+\Gamma$ ($\Gamma$ is added $n$ times).
  • We assume that $0\in \Gamma$, so that $\Gamma_n\subseteq \Gamma_{n+1}$.
  • To show that $S$ is wellordered, we have to show that any $B\subseteq S$ has a smallest element; we can assume without loss of generality that $\Gamma\cap B$ is nonempty (extending $\Gamma$ to some $\Gamma_n$ if necessary).
  • With the above assumptions, existence of a minimal element of $B$ is equivalent to the statement that the sequence $b_n:=\min (\Gamma_n\cap B)$ stabilizes.

The motivation of the question is to show that the ring of Hahn series $K((X^A))$ (with $K$ a field) is a field, but even if there's a simpler way to show it, I'm curious about the question on its own.

Thoughts? Hints?

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Do you have an example where this semigroup is not $\omega$? –  Asaf Karagila Oct 5 '12 at 21:02
    
@AsafKaragila: $\Gamma=\{(0,1),(1,0)\}\subseteq {\bf Z}^2$ with lexicographic ordering? It's still trivial, but... –  tomasz Oct 5 '12 at 21:47
    
Surely you mean the generated subgroup? I don't think this set is closed under addition. Anyway, I was talking about the framework of your assumption, namely the group itself is ordered and the semigroup is all positive elements... (this is why I wrote this semigroup and not a semigroup) –  Asaf Karagila Oct 5 '12 at 21:59
    
@AsafKaragila: well, ${\bf Z}^2$ is ordered and I meant $\Gamma$ as the generating set, as in the question. $\Gamma$ is all positive, as is the semigroup generated by it, ${\bf N}^2\setminus \{0\}$. Which part do you have a problem with? –  tomasz Oct 5 '12 at 22:14

1 Answer 1

up vote 2 down vote accepted

Let $(A,\le)$ be a totally ordered abelian group.

Lemma. If $X,Y$ are two wellordered subsets of $A$, then $X+Y$ is wellordered.

Proof. Let $B$ be a nonempty subset of $X+Y$ and assume it has no minimum. For $b\in B$ let $$x_b=\min\{x\in X\mid \exists y\in Y\colon x+y\in B, x+y<b\}$$ $$y_b=\min\{y\in Y\mid x_b+y\in B\}$$ and $s(b)=x_b+y_b$. Then $s(b)<b$. This implies $x_{s(b)}\ge x_b$, hence $y_{s(b)}<y_b$

If $b\in B$ is arbitrary, this produces an infinite descending sequence $y_b>y_{s(b)}>y_{s(s(b))}>\ldots $ in $Y$, which is impossible. Therefore $B$ must have a minimal element.$_\blacksquare$

Theorem. If $\Gamma$ a positive wellordered subset of $A$, then $\langle\Gamma\rangle$ is well-ordered.

As a wellordered set, $\Gamma$ is order-isomorphis to some ordinal $\alpha$. We proceed by induction on $\alpha$, that is: We may assume that $\langle\Gamma'\rangle$ is wellordered for all $\Gamma'$ of the form $\Gamma'=\{g\in \Gamma\mid g<\gamma\}$ for some $\gamma\in\Gamma$.

Let $B\subset\langle\Gamma\rangle$ be a nonempty set and let $b\in B$ be one of its elements. If $b=0$, it is clearly minimal in the nonnegative set $\langle \Gamma\rangle$. If $b\ne 0$, write $$b=g_1+\cdots +g_n$$ with $n\in \mathbb N$, $g_i\in \Gamma$ and let $\gamma=\max\{g_1,\ldots,g_n\}$. Then $b\le n\gamma$. Consider $b'\in B$ with $b'\le b$ and write $$b'=g'_1+\cdots +g'_m$$ with $m\in \mathbb N$, $g'_i\in \Gamma$. Then at most $n$ of the $g'_i$ are $\ge \gamma$. All other summands are in $\Gamma'=\{g\in \Gamma\mid g<\gamma\}$. By induction hypothesis, $\langle\Gamma'\rangle$ is wellordered. By the lemma, $$T:=\underbrace{(\Gamma\cup\{0\})+\cdots+(\Gamma\cup\{0\})}_n+\langle\Gamma'\rangle$$ is wellordered. We have just seen that $\{b'\in B\mid b'\le b\}\subseteq B\cap T$, hence $$\min B =\min(T\cap B).$$ $_\blacksquare$

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Yep, needs rework - I turned some inequalities –  Hagen von Eitzen Oct 5 '12 at 19:21
    
@tomasz: I should have known that showing wellorder would require the full ordinal and transfinite machinery ... –  Hagen von Eitzen Oct 5 '12 at 20:56
    
Well, maybe $<$ should be replaced with $\le$ in the def of $\preceq$. Or $0\in\Gamma$ excluded. Or do it completely different by letting $\gamma_\alpha=\min(\Gamma\setminus\langle \gamma_i\mid i<\alpha\rangle)$. I'll check which of these modification looks best. –  Hagen von Eitzen Oct 6 '12 at 8:54
    
I hope you are ok with my post-acceptance edits. I think the proof is much shortert and easier to comprehend now. –  Hagen von Eitzen Oct 6 '12 at 10:41
    
Yes, it's much more natural and easy to understand now. :) I took the liberty to fix a typo and a minor mistake at the very end, I hope you don't mind. –  tomasz Oct 6 '12 at 11:26

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