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A standard characterization of $\mathbf{R}$ uses the order and the field structure: any linearly ordered field that is archimedean and complete is isomorphic to $(\mathbf{R}, +, \times, <)$ as an ordered field.

Is there a similar characterization of $\mathbf{R}$ as an ordered group?

Is any linearly ordered group that is archimedean and complete isomorphic to $(\mathbf{R}, +, <)$ as an ordered group or is some other assumption needed? Any reference is welcome.

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  • $\begingroup$ What definition of 'archimedean' do you use? $\endgroup$ – R. van Dobben de Bruyn Jun 9 '18 at 10:06
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    $\begingroup$ The only definition of Archimedean I can imagine here is: for all $x,y>0$ there exists $n\ge 1$ such that $nx\ge y$ where $nx=x+\dots+x$ ($n$ times). $\endgroup$ – YCor Jun 9 '18 at 10:08
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    $\begingroup$ Yes, that's the standard one for linearly ordered groups. $\endgroup$ – coudy Jun 9 '18 at 10:11
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    $\begingroup$ I think that the archimedean property follows from the completeness axiom. $\endgroup$ – Liviu Nicolaescu Jun 9 '18 at 10:34
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    $\begingroup$ @LiviuNicolaescu: Thanks for the reference! I think we missunderstood each other: Zorich's book derives the Archimedean property from the completeness for linearly ordered fields. But I think the same should be true for linearly ordered groups, too. I've just found a proof for this for commutative groups, but I'm not quite sure if it's true for non-commutative groups, too. $\endgroup$ – Jochen Glueck Jun 9 '18 at 11:33
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The linearly ordered group $(\mathbb{Z},+,\le)$ is a counterexample, but that is probably not what the OP had in mind. To give a detailed description of the situation, let us use the following notation:

  • By a linearly bi-ordered group we mean a tuple $(G,\cdot,\le)$ where $(G,\cdot)$ is a group and $\le$ is a linear order on $G$ such that $ac \le bc$ and $ca \le cb$ whenever $a,b,c \in G$ such that $a \le b$. We use the notion linearly ordered group as shorthand or linearly bi-ordered group.

  • An isomorphism between two linearly ordered groups $(G,\cdot,\le)$ and $(H,\cdot,\le)$ is a group isomorphism $\varphi: (G,\cdot) \to (H,\cdot)$ such that both $\varphi$ and $\varphi^{-1}$ are increasing.

  • A linearly ordered group $(G,\cdot,\le)$ (whose neutral element we denote by $e$) is called Archimedean if, for all $a,b > e$ there exists an integer $n \in \mathbb{N}$ such that $a^n \ge b$.

  • We call a linear order an a set $S$ complete if every non-empty subset of $S$ that is bounded above has a supremum in $S$ (equivalently, every non-empty subset of $S$ that is bounded below has an infimum in $S$). Note that this property is sometimes called conditionally complete (instead of complete) in the literature.

Theorem 1. Let $(G,\cdot,\le)$ be an Archimedean linearly ordered group. Then $(G,\cdot)$ is isomorphic to an ordered subgroup of $(\mathbb{R},+,\le)$ (i.e. a subgroup of $(\mathbb{R},+)$ which carries the order inherited from $\mathbb{R}$). In particular, $(G,\cdot)$ is commutative.

This result can, for instance, be found in Theorem 1 in Section IV.1 of

Fuchs, L., Partially ordered algebraic systems, Oxford-London-New York-Paris: Pergamon Press. IX, 229 p. (1963). ZBL0137.02001. There, the theorem is attributed to Hölder.

As kindly pointed out by user Alec Rhea in the comments, there is a related result by Hahn which gives a description of all commutative linearly ordered groups.

Next we note that linearly ordered groups whose order is complete are automatically Archimedean:

Theorem 2. Let $(G,\cdot,\le)$ be a linearly ordered group and assume that the order $\le$ on $G$ is complete. Then $(G,\cdot,\le)$ is Archimedean.

Proof. Let $e$ denote the neutral element of $(G,\cdot)$, let $a,b > e$ and assume for a contradiction that $a^n < b$ for all $n \in \mathbb{N}$. Then the set $S := \{a^n: \, n \in \mathbb{N}\}$ has a supremum $s$ in $G$. We have $a^{-1}s < s$, so $a^{-1}s < a^n$ for some $n \in \mathbb{N}$. Consequently, $s < a^{n+1} \le s$, which is a contradiction.

In an earlier version of this post, a more complicated proof of Theorem 2 was given. The above version of the proof was kindly pointed out by user Emil Jeřábek in the comments.

Remark 3. Note that the existence of inverse elements is esssential not only for the proof, but also for the validity of Theorem 2. Indeed, the set $[0,\infty) \times [0,\infty)$, endowed with componentwise addition and the lexicographical order, is an example of a linearly ordered semigroup (whose composition operation is strictly monotone in both components) which is order complete but not Archimedean.

By combining Theorems 1 and 2 we arrive at the following corollary which, I think, answers the question of the OP:

Corollary 4. Let $(G,\cdot,\le)$ be a linearly ordered group. If the order $\le$ on $G$ is complete, then $(G,\cdot,\le)$ is isomorphic to one of the three linearly ordered groups $(\{0\},+,\le)$, $(\mathbb{Z},+,\le)$ and $(\mathbb{R},+,\le)$.

Proof. According to Theorem 2 $(G,\cdot,\le)$ is Archimedean, so it is isomorphic to an ordered subgroup $(H,+,\le)$ of $(\mathbb{R},+,\le)$ due to Theorem 1. If $H$ has only one element, then obviously $H = \{0\}$, so assume that $H$ has at least two elements. Now we distinguish two cases:

First case: $h_0 := \inf \{h \in H: \, h > 0\} > 0$. Then it is easy to see that $H = h_0 \mathbb{Z}$, so $(H,+,\le)$ is isomorphic to $(\mathbb{Z},+,\le)$.

Second case: $\inf \{h \in H: \, h > 0\} = 0$. Then one readily checks that the set $H$ is dense in $\mathbb{R}$, and the completeness of the order on $H$ implies that $H = \mathbb{R}$. This proves the corollary.

-- Note on edits made 2018-06-10. -- I rewrote the answer and consolidated the various edits from the previous versions in order to make this post better readable for future visitors. I also incorporated various suggestions by users Alec Rhea, Emil Jeřábek and YCor, so let me thank them for their comments!

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    $\begingroup$ Very nice answer, it may be worth mentioning that the embedding theorem attributed to Holder at the beginning is a special case of the Hahn embedding theorem (en.wikipedia.org/wiki/Hahn_embedding_theorem), which offers a nice characterization of all totally ordered commutative groups. $\endgroup$ – Alec Rhea Jun 9 '18 at 22:39
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    $\begingroup$ By the way if you don't assume $G$ commutative you'd need to clarify whether "ordered group" means the order is left-invariant or both left and right invariant. Both convention exist, so it's better write left-ordered group and bi-ordered group. $\endgroup$ – YCor Jun 9 '18 at 23:44
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    $\begingroup$ The proof of Theorem 6 is unnecessarily complicated. Let $a>1$. If $\{ a^n:n\in\mathbb N\}$ is not cofinal in $G$, it has a least upper bound $s$. Since $a^{-1}s<s$, we have $a^{-1}s\le a^n$ for some $n$. But then $s\le a^{n+1}<a^{n+2}\le s$, a contradiction. $\endgroup$ – Emil Jeřábek Jun 10 '18 at 7:21
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    $\begingroup$ Corollary 4 is very closely related to the classification of factors (von Neumann algebras with trivial center) into type I, type II, and type III. $\endgroup$ – André Henriques Jun 10 '18 at 20:16
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    $\begingroup$ Let $R$ be a factor. Then the collection of all isomorphism classes of $R$-modules under the operation of direct sum is isomorphic to either $(\mathbb N\cup\{\infty\},+)$ (the factor is of type I), or $(\mathbb R_{\ge 0}\cup\{\infty\},+)$ (the factor is of type II), or $(\{0\}\cup\{\infty\},+)$ (the factor is of type III). Note that I'm over-simplifying things a tiny bit by not distinguishing between different sizes (cardinalities) of $\infty$. $\endgroup$ – André Henriques Jun 11 '18 at 19:53

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