Questions tagged [ordered-groups]

Groups (possibly semigroups) endowed with possibly left/right/bi-invariant partial/total orderings. Study of such orders on groups.

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3
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1answer
115 views

Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid?

Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not ...
4
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2answers
426 views

Unique product group which is not right orderable

(1) I am looking for an example of a u.p (unique product) group which is not right orderable (RO). Almost any group I pick up (obviously torsion-free, as u.p. group cannot have nontrivial torsion ...
3
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0answers
92 views

Partial orders on $\mathbb{N}^m$ compatible with addition

I'm looking for a classification (or just non-trivial examples) of partial orders on monoid $\mathbb{N}^{m}$ that are compatible with addition. That is, partial orders $\leq$ satisfying two additional ...
6
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1answer
262 views

Groups with three conjugacy classes that define an ordering

Consider the following property for a group $(\mathcal{G},\cdot,1)$: There are exactly three conjugacy classes $\{1\}$, $\mathcal{C}_1$, $\mathcal{C}_2$ in $\mathcal{G}$, and we have $\mathcal{C}_1 \...
26
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4answers
2k views

The sum of two well-ordered subsets is well-ordered

Apologies if the answer is trivial, this is far from my domain. In order to define the field of Hahn series, one needs the following fact: if $A,B$ are two well-ordered subsets of $\mathbb{R}$ (or any ...
7
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2answers
411 views

A linearly orderable monoid which does not embed into a linearly orderable group

It is known (after an example of A.I. Mal'cev) that there exist cancellative semigroups which do not embed into a group. On the other hand, it is not difficult to see that every linearly orderable ...
19
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1answer
552 views

Is Thompson's group definably orderable?

Is Thompson's group $F$ definably left-orderable? definably bi-orderable? Orderability definitions: Recall that a group $G$ is left-orderable (resp. bi-orderable) if it admits a left-invariant (resp. ...
3
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0answers
68 views

Extending a representation of a free group to an extension of a mapping torus

Given a free group on $n$ generators, $F_n$, $\phi$ an automorphism of $F_n$, and a non-trivial representation $\rho: F_n \rightarrow \operatorname{Homeo}_+(\mathbb{R})$, are necessary and sufficient ...
11
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1answer
516 views

Partial word orders on groups

This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial word order if for every $a\le b\le c$ we have $|...
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0answers
62 views

Compatibility with multiplication of a cyclic order on a ring

I am copying my question from here: https://math.stackexchange.com/q/3233462/427611. Is it correct that $\mathbb Z/3\mathbb Z$ and $\mathbb Z/4\mathbb Z$ are the only rings with three or more ...
4
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0answers
273 views

What is known about orbifolding ordered groups and sets? Who has been involved? Links to Lee metrics?

In mathematical music theory several ordered groups are considered. Some examples contain the frequency space or Tonnetzes. Other groups (commutative and non-commutative ones) are discussed by Dawid ...
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1answer
124 views

Generating totally ordered free commutative monoids

Let’s say I have a set $A$. I build the free commutative monoid $M$ generated by $A$. When can a well-order on $A$ be extended to $M$, in a way that is compatible with its monoid structure? I am ...
0
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1answer
182 views

Existence of an orbit of exponential growth for group acting on the real line

Let G be a non-abelian finitely generated subgroup of increasing homeomorphisms of the real line having a fixed point free element $h$ ($hx>x$ for all $x$ in the line). Is there a real number $a$ ...
9
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1answer
334 views

Left orderable linear groups

Are all torsion-free finitely generated linear groups over $\mathbb{C}$ left orderable? In particular, are torsion-free congruence subgroups of $SL_n(\mathbb{Z})$ left orderable?
4
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4answers
1k views

Why do we choose the standard total order on the integers?

I understand why the set of natural numbers $\mathbb N = \{ 0, 1, 2, \cdots \}$ is equipped with a total order. Indeed, every monoid has a pre-order, where $$n' \succeq n \quad \mathrm{if~and~only~if} ...
4
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3answers
751 views

Group action on the real line

I was wondering about the following question: if you have a faithful action of a group $G$ on the real line $\mathbb{R}$ by orientation-preserving homeomorphisms, it is easy to construct a new action ...
3
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1answer
146 views

Compatible total orderings of the group $\mathbb{Z}^\mathbb{N}$

Given the additive group of the module $\mathbb{Z}^\mathbb{N}$ and a total ordering of the group that is compatible with addition and where $\chi_{\{n\}} > 0$ for all $n \in \mathbb{N}$, can we say ...
3
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0answers
574 views

Braided lobsters

If $(X,m)$ is a median algebra, then for each $x\in X$, define an operation $\wedge_{x}$ by letting $y\wedge_{x}z=m(x,y,z)$. Then $(X,\wedge_{x})$ is a meet-semilattice with least element $x$. Define ...
6
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2answers
349 views

orders and length functions on finitely generated groups

Let $G$ be a finitely generated group with the natural word length function ($|x|$ is the length of the shortest word in generators of $G$ representing $x$). We call a partial left invariant order $\...
15
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3answers
947 views

Can Suslin (or Aronszajn) lines ever be orderings of abelian groups?

I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way? Let $\mathcal{C}$ be a class of (...
5
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0answers
498 views

Unique product groups (and semigroups)

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a ...
3
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1answer
195 views

Extensions of partial orders to linear orders on (nonabelian) groups

If $G$ is a group with a (left) linear order, does every (left) partial order on $G$ extend to a (left) linear order? The answer is affirmative on abelian groups, where being torsion-free is ...
4
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0answers
303 views

Amenable groups acting on the real line, that are not subexponentially-amenable

In the literature, there are several examples of solvable groups acting faithfully by order-preserving homeomorphisms of the real line. There are also examples of groups of intermediate growth with ...
7
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2answers
287 views

Residual $p$-finiteness of principal congruence subgroups

Let $\Gamma(N)$ be the principal congruence subgroup of level $N$ in $\mathrm{SL}_n(\mathbf{Z})$, where $n\geq 3$. Then $\Gamma(N)$ is residually $p$-finite for all primes $p$ dividing $N$. Can $\...
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1answer
305 views

Constructing a convex valuation ring/ordered group of rank $n$

I know at least one method of constructing a convex valuation ring of rank $n$ (but it is rather complicated). What are the easiest methods of doing this? Given a natural number $n$ I want to have a ...
0
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2answers
187 views

Direct limit of lattice-ordered groups

In general, any abelian group can be expressed as a direct limit of its f.g. subgroups. For the case of $\ell$-group (lattice-ordered group) is that true or not? As an abelian group we do not have ...
12
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1answer
377 views

Bi-orderability of Baumslag-Solitar group $\langle a,b \mid a^{-1} b^m a = b^n\rangle$ and of $\langle a,b \mid a^{-1} b a^m = b^n\rangle$

We say that a group $(A, \cdot)$ is bi-orderable if there exists a total order $\preceq$ on $A$ such that $xz \prec yz$ and $zx \prec zy$ for all $x,y,z \in A$ with $x \prec y$. Let $m,n$ be non-zero ...
4
votes
1answer
180 views

Totally right preorderable groups

Are there any known non-trivial sufficient conditions, or full characterizations, of a totally right-preorderable group? More precisely: totally right-preorderable: has a non-trivial total right-...
3
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1answer
268 views

For which groups is (non-)left orderability decidable?

Mainly, my question is in the title, but let me be more precise here. Let $G$ be a finitely presented group with solvable word problem. If G is not left-orderable, is there an finite-time algorithm ...
8
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1answer
288 views

Is there a left-orderable profinite group?

Is there a nontrivial profinite group $G$ with a binary transitive relation $<$ such that $x<y$ implies $x\neq y$, and for any different $x,y \in G$ either $x < y$ or $y < x$ and such ...
3
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2answers
934 views

Every abelian torsion-free group is strictly totally orderable (via the compactness theorem)

Let $\mathbb G = (G, +)$ be a group. We say that $\mathbb G$ is strictly totally orderable (others would say bi-orderable) if there exists a total order $\preceq$ on $G$ such that $x+z \prec y + z$ ...
2
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1answer
184 views

Is there a non-right-orderable torsion-free quotient group of the braid group on 3 strands?

The braid group on 3 strands has the presentation $\langle x,y \;|\; xyx=yxy\rangle$. A group $G$ is called right-orderable if there is a total order $<$ on the set $G$ such that if $a<b$ then $...
1
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1answer
117 views

Orderable subgroup of the braid groups over the 2-sphere

$$B_{n}(S^2)=\langle \sigma_1,\sigma_2,...\sigma_{n-1}\mid \sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i} \text{ if } |i-j|>1;\qquad$$ $$\qquad \sigma_{i}\sigma_{j}\sigma_{i}=\sigma_{j}\sigma_{i}\sigma_{...
4
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1answer
347 views

Do all right orderable groups have the Haagerup property?

Do all right orderable groups have the Haagerup property? Recall that a group is right orderable if there exists a total order $\leq$ on it such that $a\leq b\Rightarrow ac\leq bc$. This property is ...
2
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2answers
434 views

Non-archimedean group over the reals

I have a totally ordered group $(\mathbb{R};\leq,\oplus,0,-)$ with the reals as base set satisfying monotonicity, i.e. for all $x,y,z$ we have that if $y\leq z$ then $x\oplus y \leq x\oplus z$, and I ...
4
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1answer
126 views

Characterization of Archimedean linearly ordered monoids

In this question, it is shown that all Archimedean ordered groups are isomorphic to an ordered subgroup of $\mathbb R$. Additionally, it is shown that if such a group is complete, then it is ...
2
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1answer
206 views

Example involving partially ordered Abelian groups

Definition 1: Let $(G,\leq)$ be a nonzero partially ordered Abelian group with order unit $u$. (Recall that $u\in G$ is a order unit if, for every $g\in G$, there exists $N\in\mathbb N$ such that $-Nu\...
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0answers
61 views

Pure (ordered) subgroups

Let $H,G$ be abelian groups with $H \leq G$. We say that $H$ is a pure subgroup of $G$ if for every $n \in \mathbb N$ and $h \in H$ the following holds: If $h$ is $n$-divisible in $G$, then $h$ is $n$-...
5
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1answer
253 views

Extending homomorphisms between ordered abelian groups

Let $\Omega$ be a linearly (i.e. fully) ordered set, and let $\Lambda_{\Omega}$ be the ordered abelian group consisting of those $(\lambda_\omega)_{\omega\in\Omega}\in\mathbb{R}^{\Omega}$ with well-...
2
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0answers
403 views

An equivariant Hahn embedding theorem?

The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group ...
0
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1answer
167 views

Spliting of short exact exact sequences of partially ordered groups

Consider a short exact sequence of partially ordered groups $$0 \longrightarrow H \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} G/H \longrightarrow 0 ,$$ where $H$ is a ...
4
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0answers
80 views

Is there a name for this kind of structure? (Not quite a lattice-ordered group)

I'm looking at a certain class of groups $G$ that come with a partial order $\le$ on the elements. So far it looks like $(G,\le)$ has the following properties: The partial order is invariant under ...
8
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1answer
978 views

Lattice-ordered commutative monoids

By a lattice-ordered monoid, I mean a structure $(A,0,{+},{\vee},{\wedge})$ such that $(A,0,{+})$ is a (not necessarily commutative) monoid, $(A,{\vee},{\wedge})$ is a lattice, and the two ...
1
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1answer
367 views

Is $x + y \ne y+nx$ for $x \ne 0$ and $n \ge 2$ (in an ordered group)?

Let $(A, +, \preceq)$ be an ordered group, namely $(A, +)$ is a group and $\preceq$ is a total order on $A$ such that $x + z \prec y + z$ and $z + x \prec z + y$ for all $x,y,z \in A$ with $x \prec y$....
2
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2answers
271 views

Conditions for a group to be lattice-ordered

Given a set $S$ with a group operation $\cdot$ and a lattice ordering $\leq$, I wish to know when we can say that $\cdot$ preserves $\leq$, i.e. $(x\vee y)z=xz\vee yz$ and similarly for meets. ...
2
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1answer
371 views

Ordered groups - examples

Let $G=\operatorname{BS}(m,n)$ denote the Baumslag–Solitar group defined by the presentation $\langle a,b: b^m a=a b^n\rangle$. We assume that $G$ is non-abelian, i.e., $m,n\in\mathbb{Z}\...
32
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6answers
4k views

What's a non-abelian totally ordered group?

Because I have heard the phrase "totally ordered abelian group", I imagine there should be non-abelian ones. By this I mean a group with a total ordering (not to be confused with a well-ordering) ...
3
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0answers
77 views

A Krull-Schmidt theorem for partially ordered groups

If $G$ is a po-group (ie. partially ordered group), we say that $G$ is po-indecomposable if it's not the direct product of two non trivial subgroups (such subgroups are necessary convex and normal). ...
5
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0answers
246 views

How can you order a free group?

A left order on a (discrete) group $G$ is a total order on $G$ satisfying $\forall g,h,k \in G: g < h \implies kg < kh$. A right order is defined symmetrically, and a biorder is an order that is ...
1
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1answer
84 views

Name of the class of linearly ordered groups with no minimal positive element

Is there a special name for a linearly ordered group $G$ such that for every positive element $g\in G$ there exists an element $h\in G$ such that $e<h<g$?