partial order | Samin Riasat

Let $a$ and $b$ be integers. Consider the ring

$G=\{ax+by:x,y\in\mathbb Z\}$ .

This is a well-ordered group. So by a result in this post it is infinite cyclic. We call the positive generator $(a,b)$ the greatest common divisor (GCD) of $a$ and $b$ .

Consider now the ring

$L=\{m\in\mathbb Z:m$ is divisible by both $a$ and $b\}$ .

It is also a well-ordered group. Hence it is generated by a single positive element $[a,b]$ , called the least common multiple (LCM) of $a$ and $b$ .

Let

$M_a=\{ax:x\in\mathbb Z\}$ and $M_b=\{by:y\in\mathbb Z\}$ .

$M_a$ , $M_b$ and $L$ are subings of $G$ . Moreover, $G=M_a+M_b$ and $L=M_a\cap M_b$ . So by the Chinese Remainder Theorem,

$G/L\cong G/M_a\times G/M_b$ ,

which can be written as

$(\mathbb Z/L)/(\mathbb Z/G)\cong(\mathbb Z/M_a)/(\mathbb Z/G)\times(\mathbb Z/M_b)/(\mathbb Z/G)$

by the third isomorphism theorem. The groups in brackets are all finite groups of orders $[a,b]$ , $(a,b)$ , $a$ and $b$ . Hence $[a,b]/(a,b)=ab/(a,b)^2$ , i.e.,

$\boxed{ab=(a,b)[a,b]}$ .

In general, let $a_1,\dots,a_n\in\mathbb Z$ . As before, we can define

$G=\{a_1x_1+\cdots+a_nx_n:x_i\in\mathbb Z\ \forall i\}$

$L=\{m\in\mathbb Z:m$ is divisible by $a_i\ \forall i\}$

$M_i=\{a_ix:x\in\mathbb Z\}$ for $i=1,\dots,n$ .

Then $G=M_1+\cdots+M_n$ and $L=M_1\cap\cdots\cap M_n$ . As before,

$G/L\cong\displaystyle\prod_{i=1}^n(G/M_i)$ ,

i.e.,

$(*)\qquad\qquad\qquad (\mathbb Z/L)/(\mathbb Z/G)\cong\displaystyle\prod_{i=1}^n((\mathbb Z/M_i)/(\mathbb Z/G))$ .

So $\displaystyle [a_1,\dots,a_n]/(a_1,\dots,a_n)=a_1\cdots a_n/(a_1,\dots,a_n)^n$ , i.e.

$\boxed{a_1\cdots a_n=(a_1,\dots,a_n)^{n-1}[a_1,\dots,a_n]}$ .

If we replace $\mathbb Z$ by $\mathcal O_K$ for any number field $K$ , then $(*)$ takes the form

$(\mathcal O_K/L)/(\mathcal O_K/G)\cong\displaystyle\prod_{i=1}^n((\mathcal O_K/M_i)/(\mathcal O_K/G))$ .

Taking cardinalities gives the following equation in terms of ideal norms

$N(L)/N(G)=\displaystyle\prod_{i=1}^n(N(M_i)/N(G))$ .

Thus

$\boxed{N_{K/\mathbb Q}(a_1\cdots a_n)=N(G)^{n-1}N(L)}$ .

This idea first came to my mind a long time ago when I was trying to generalise the application of the division algorithm. All I came up with is the following:

Proposition 0. Let $S$ be a subgroup of $(\mathbb R,+)$ . If $S$ has a least positive element, then $S$ is cyclic. Otherwise $S$ has arbitrarily small positive elements.

A few days ago, in a Number Theory lecture, we proved that the minimal solution to Pell’s equation generates all solutions. It reminded me of the division algorithm again, so it was time again to try to generalise this, because now I have a few more tools in hand!

Let’s say $S$ is a set where the division algorithm applies. We definitely need some sort of partial order in $S$ to say that the ‘remainder’ must be ‘less than’ the ‘divisor’. We might want $S$ to be closed under some operation (so that we can repeatedly ‘subtract’ the ‘divisor’ from the ‘dividend’) and we also need an inverse operation (i.e. the ‘subtraction’).

So first of all, we need $S$ to be a poset. In addition, the closure and inverse operations suggest that we want $S$ to be a group. How should the order behave under the group operation? Clearly we want it to be compatible with the operation. We also probably want the inverse of a ‘positive’ element to be ‘negative’, and vice-versa. Do we need the group to be abelian? Maybe, but let’s not impose that condition yet.

So to put our ideas into work, let $(G,+)$ be a group with a partial order $\le$ such that for all $g,g_1,g_2\in G, g_1\le g_2\Rightarrow g+g_1\le g+g_2$ and $g_1+g\le g_2+g$ . We say that an element $g\in G$ is positive if $0\le g$ , where $0$ is the identity element of $G$ . Define the positive cone of $G$ to be the set $G^+:=\{g\in G:0\le g\}$ of all positive elements.

Okay now we hopefully have all the necessary definitions in place. Let’s see if we can prove anything using these. The first thing that we want is probably: if $0\le g$ , then $-g\le 0$ . This follows easily: $0\le g$ , add $-g$ to both sides and we are done. It works the other way around as well, so in fact we have proved:

Lemma 1. $0\le g\Leftrightarrow -g\le 0$ .

That was good. We wanted the inverse of positive elements to be negative and it just followed from the definition. But we want more! So take a non-zero element $g\in G$ . By Lemma 1 we can take $g$ to be positive without loss of generality. How about adding something to both sides of $0\le g$ again? Last time we added $-g$ . We can add $0$ , but that doesn’t change anything. So the only obvious choice left is to add $g$ : $g\le g+g=2g$ . Now what? Let’s add $g$ again! $g+g\le 2g+g$ , i.e. $2g\le 3g$ . Combining the last two gives $0\le g\le 2g\le 3g$ . It follows by induction that $mg\le ng$ for all integers $0\le m\le n$ (note: here $ng=g+\dots+g$ , $n$ times). This looks promising.

What about ‘negative’ elements? Note that by Lemma 1, $-g\le 0$ . Adding $-g$ to both sides yields $-2g\le -g$ , and so on. So we have another nice result:

Lemma 2. $mg\le ng$ for all integers $m\le n$ and $g\in G^+$ .

It seems that this is all we can derive from our first principles. So let’s apply more restrictions on $G$ . Let $a,b\in G$ be positive with $b\le a$ . As in the division algorithm, let’s look at $a-b,a-2b,\dots$ etc. We want this sequence to stop as soon as $a-nb$ becomes negative. How do we do this? In other words, we want the set $\{a-nb:n\in\mathbb Z\}$ to have a least positive element. Did something just pop up in your mind? A set having a least element must have reminded you of something like… the well-ordering principle! So how about we impose the extra condition that $\le$ is a well-order on $G$ ? Well, that’s clearly absurd, because for any positive $g$ the set $\{ng:n\in\mathbb Z\}$ has no least element. How about least positive element then? In other words, let’s say $G^+$ is well-ordered under $\le$ .

Now $G$ has quite a few nice properties: it is a group under $+$ , $\le$ is an order on $G$ preserving $+$ , and its positive cone is well-ordered. Let’s see if our ideas work now.

Let $\varepsilon$ be the least non-zero element in $G^+$ and $g\in G$ be any non-zero element. Without loss of generality, $0<g$ . Then $g\in G^+$ so $\varepsilon\le g$ . Consider the elements $n\varepsilon$ for $n\in\mathbb Z$ . We want $n\varepsilon\le g<(n+1)\varepsilon$ for some $n$ . Can we achieve this? We certainly have $n\varepsilon\le g$ for $n=1$ , so we need $g<n'\varepsilon$ for some $n'$ . By Lemma 2 $n'$ must be greater than $n$ . How do we know that $n'$ exists?

Suppose it doesn’t. Then $n\varepsilon\le g$ for all $n\in\mathbb Z$ by totality (recall that a well-order is a total order) and Lemma 2. Then $0\le g+n\varepsilon\;\forall n$ , so $g+n\varepsilon\in G^+\;\forall n$ . Hence $\{g+n\varepsilon:n\in\mathbb Z\}\subset G^+$ , so it has a least element $g+m\varepsilon$ . Then $g+m\varepsilon\le g+n\varepsilon\;\forall n$ which implies $0\le n\varepsilon$ for all $n\in\mathbb Z$ , a contradiction.

That was really good! Now we can take the maximal $n$ such that $n\varepsilon\le g$ . Then $g<(n+1)\varepsilon$ . Then $0\le g-n\varepsilon<\varepsilon$ ; the left inequality says $g-n\varepsilon\in G^+$ , and the right inequality says $g-n\varepsilon<\varepsilon$ . So $g-n\varepsilon=0$ and $g=n\varepsilon$ . This is exactly what we wanted.

We have shown that $G=\langle\varepsilon\rangle$ . In fact we can do more. Clearly $G$ cannot be finite. Because otherwise $\varepsilon$ must have finite order, i.e. $k\varepsilon=0$ for some positive integer $k$ . Then $0\le\varepsilon\le 2\varepsilon\le\dots\le k\varepsilon=0$ by Lemma 2. So all of these must be equalities (by antisymmetry), i.e. $\varepsilon=0$ , a contradiction.

So our restrictions have not only worked, we’ve shown that all groups with these properties essentially have the same structure, that of the infinite cyclic group. Let’s give $G$ a name: we say that the group $G$ is well-ordered if the set $G^+$ is well-ordered under $\le$ . We have thus proved:

Proposition 1. The only non-trivial well-ordered group is the group $(\mathbb Z,+)$ of integers (up to isomorphism).

Now we can give one-line proofs of the following facts using our Proposition 1: (here any ordering is under the usual $\le$ order in $\mathbb R$ )

Corollary 1. The $\gcd$ of two natural numbers exists, and is their least positive linear combination.

Proof. For $a,b\in\mathbb N$ , the additive group $G=\{ax+by:x,y\in\mathbb Z\}$ is well-ordered, and so is equal to $\langle\varepsilon\rangle$ for $\varepsilon$ the least positive element of $G$ . $\square$

Corollary 2. $\mathbb Z$ is a PID.

Proof. Any ideal in $\mathbb Z$ is a well-ordered group, and so must be $\langle d\rangle$ for some $d$ . $\square$

Corollary 3. If $x_0+y_0\sqrt d$ is the least solution $>1$ to Pell’s equation $x^2-dy^2=1$ , then all solutions are given by $x_n+y_n\sqrt d=(x_0+y_0\sqrt d)^n$ for $n\in\mathbb Z$ .

Proof. The solutions $x_n+y_n\sqrt d$ to Pell’s equation form a subgroup of the multiplicative group of units in the ring $\mathbb Z[\sqrt d]$ . Since it is well-ordered, the conclusion follows. $\square$

We can even improve Proposition 0 a little:

Corollary 4. If $S$ is a subgroup of $(\mathbb R,+)$ , then the following are equivalent:

(i) $S$ is well-ordered;

(ii) $S$ is not dense;

(iii) $S$ is cyclic.

Proof. $(i)\Leftrightarrow (iii)$ by Proposition 1. $(iii)\Rightarrow (ii)$ is clear. Suppose that $S$ is not dense. Then there exist $a,b\in S$ with $a<b$ such that $(a,b)\cap S=\emptyset$ . Then $(0,b-a)\cap S=\emptyset$ , because $x\in (0,b-a)\cap S\Rightarrow a+x\in (a,b)\cap S$ . Therefore $b-a$ is the minimal element in $S^+\backslash\{0\}$ , i.e. $S=\langle b-a\rangle$ . Thus $(ii)\Rightarrow (iii)$ . $\square$

A consequence of Corollary 4 is:

Corollary 5. Let $G$ be a group. If $G$ has a faithful one-dimensional real representation $\rho: G\to\mathbb R^\times$ , then $\rho(G)$ is dense if and only if $G\not\cong\mathbb Z/\mathbb Z,\mathbb Z/2\mathbb Z, \mathbb Z$ .

Note that $\rho(G)$ is dense if and only if it has two $\mathbb Z$ –linearly independent elements; therefore $G$ must be torsion-free. And since $\rho$ is faithful, $G$ must be abelian: $\rho(ghg^{-1}h^{-1})=1\Rightarrow ghg^{-1}h^{-1}=e$ , i.e. $gh=hg$ for all $g,h\in G$ . Thus we get the following nice result: