Orders on locally compact abelian groups

An order $P$ on $\Gamma$ is a subset that satisfies the three axioms: $P+P\subset P$; $P\cup (-P)=\Gamma$; and $P\cap (-P)-\{0\}$. We recall from [1] the following property of orders.

Theorem 2.1   Let $P$ be a measurable order on $\Gamma$. There are a totally ordered set $\Pi$ with largest element $\alpha_0$; a chain of subgroups $\{C_\alpha\}_{\alpha\in\Pi}$ of $\Gamma$; and a collection of continuous real-valued homomorphisms $\{\psi_\alpha\}_{\alpha\in\Pi}$ on $\Gamma$ such that:
(i) for each $\alpha\in\Pi$, $C_\alpha$ is an open subgroup of $\Gamma$;
(ii) $C_\alpha\subset C_\beta$ if $\alpha > \beta$.
Let $D_\alpha=\{\chi\in C_\alpha:\ \psi_\alpha(\chi)=0\}$. Then, for every $\alpha\in\Pi$,
(iii) $\psi_\alpha(\chi)>0$ for every $\chi\in P\cap (C_\alpha\setminus D_\alpha)$,
(iv) $\psi_\alpha(\chi)<0$ for every $\chi\in (-P)\cap (C_\alpha\setminus D_\alpha).$
(v) When $\Gamma$ is discrete, $C_{\alpha_0}=\{0\}$; and when $\Gamma$ is not discrete, $D_{\alpha_0}$ has empty interior and is locally null.

When $\Gamma$ is discrete, Theorem 2.1 can be deduced from the proof of Hahn's Embedding Theorem for orders (see [13, Theorem 16, p.59]). The general case treated in Theorem 2.1 accounts for the measure theoretic aspect of orders. The proof is based on the study of orders of Hewitt and Koshi [18].

For $\alpha\in\Pi$ with $\alpha\neq\alpha_0$, let

$$S_\alpha\equiv P\cap (C_\alpha \setminus D_\alpha) = \left\{ \chi\in C_\alpha\setminus D_\alpha:\ \ \psi_\alpha(\chi)\geq 0\right\}$$ (11)

$$= \left\{ \chi\in C_\alpha :\ \ \psi_\alpha(\chi)> 0\right\}.$$ (12)

For $\alpha=\alpha_0$, set

$$S_{\alpha_0}=\left\{ \chi\in C_{\alpha_0}:\ \ \psi_{\alpha_0}(\chi)\geq 0\right\}.$$ (13)

Note that when $\Gamma$ is discrete, $C_{\alpha_0}=\{0\}$, and so $S_{\alpha_0}=\{0\}$ in this case.

If $A$ is a subset of a topological space, we will use $\overline{A}$ and $A^\circ$ to denote the closure, respectively, the interior of $A$.

Remarks 2.2   (a) It is a classical fact that a group $\Gamma$ can be ordered if and only if it is torsion-free. Also, an order on $\Gamma$ is any maximal positively linearly independent set. Thus, orders abound in torsion-free abelian groups, as they can be constructed using Zorn's Lemma to obtain a maximal positively linearly independent set. (See [18, Section 2].) However, if we ask for measurable orders, then we are restricted in many ways in the choices of $P$ and also the topology on $\Gamma$. As shown in [18], any measurable order on $\Gamma$ has nonempty interior. Thus, for example, while there are infinitely many orders on $\Bbb R$, only two are Lebesgue measurable: $P=[0,\infty [$, and $P=]-\infty,0]$. It is also shown in [18, Theorem (3.2)] that any order on an infinite compact torsion-free abelian group is non-Haar measurable. This effectively shows that if $\Gamma$ contains a Haar-measurable order $P$, and we use the structure theorem for locally compact abelian groups to write $\Gamma$ as $\Bbb R^a\times \Delta$, where $\Delta$ contains a compact open subgroup [20, Theorem (24.30)], then either $a$ is a positive integer, or $\Gamma$ is discrete. (See [1].)
(b) The subgroups $(C_\alpha)$ are characterized as being the principal convex subgroups in $\Gamma$ and for each $\alpha\in\Pi$, we have

$$D_\alpha=\bigcup_{\beta>\alpha}C_\beta.$$

Consequently, we have $C_\alpha\subset D_\beta$ if $\beta <\alpha$. By construction, the sets $C_\alpha$ are open. For $\alpha< \alpha_0$, the subgroup $D_\alpha$ has nonempty interior, since it contains $C_\beta$, with $\alpha<\beta$. Hence for $\alpha\neq\alpha_0$, $D_\alpha$ is open and closed. Consequently, for $\alpha\neq\alpha_0$, $C_\alpha\setminus D_\alpha$ is open and closed.

(c) Let $\psi:\ \Gamma_1\rightarrow \Gamma_2$ be a continuous homomorphism between two ordered groups. We say that $\psi$ is order-preserving if $\psi(P_1)\subset P_2$. Consequently, if $\psi$ is continuous and order preserving, then $\psi(\overline{P_1})\subset \overline{P_2}$.

For each $\alpha\in\Pi$, let $\pi_\alpha$ denote the quotient homomorphism $\Gamma\rightarrow \Gamma/C_\alpha$. Because $C_\alpha$ is a principal subgroup, we can define an order on $\Gamma/C_\alpha$ by setting $\psi_\alpha(\chi)\geq 0\Longleftrightarrow \chi\geq 0$. Moreover, the principal convex subgroups in $\Gamma/C_\alpha$ are precisely the images by $\pi_\alpha$ of the principal convex subgroups of $\Gamma$ containing $C_\alpha$. (See [1, Section 2].)

We end this section with a useful property of orders.

Proposition 2.3   Let $P$ be a measurable order on $\Gamma$. Then $\overline{P}$ is a ${\cal T}$-set.

Proof.    If $\Gamma$ is discrete, there is nothing to prove. If $\Gamma$ is not discrete, the subgroup $C_{\alpha_0}$ is open and nonempty. Hence the set $C_{\alpha_0}\cap \{\chi\in \Gamma:\psi_{\alpha_0} (\chi)>0\}$ is nonempty, with 0 as a limit point. Given an open nonempty neighborhood $U$ of 0, let

$$W=U\cap C_{\alpha_0}\cap \{\chi\in \Gamma:\psi_{\alpha_0} (\chi)>0\}.$$

Then $W$ is a nonempty subset of $U\cap P$. Moreover, it is easy to see that $W+\overline{P}\subset P\subset \overline{P}$, and hence $\overline{P}$ is a ${\cal T}$-set.