# Questions tagged [ordered-groups]

An ordered group is a group with a (partial) order which the group operation preserves.

61 questions
Filter by
Sorted by
Tagged with
158 views

• 273
38 views

### ordered topological groups

Is there a partially order relation on $S^1$ whose corresponding order topology coincides to the standard topology of circle?Is there a totally order relation with such property? what is a theory ...
• 1,730
115 views

### On convex subgroups of totally ordered groups.

I am trying to find an example of a totally ordered abelian group $(\Gamma,+)$ such that the set of its convex subgroups is $\textbf{not}$ totally ordered. I think there should be such an example, ...
1 vote
47 views

### Does the Unique Product Property mean that there are only two elements in the set that can create any other in the set, given an operation?

I was under the impression that the way it's defined in the title was the correct way to interpret the UPP(as that was what I was searching for) from the definition here: https://mathoverflow.net/...
1 vote
78 views

### Coproduct in the Category of Totally Ordered Groups

This is a question that arised from a problem in valuation theory of commutative rings. There I need to construct a totally ordered Abelian group "containing" every member of a given family of totally ...
• 1,290
1 vote
42 views

### Property of ordered groups.

A totally ordered abelian group is an abelian group (G,+) with a total order $\leq$ such that for all $a,b,c \in G$ if $a \leq b$ then $a+c \leq b+c$. We will say that an ordered abelian group is ...
• 2,909
220 views

### Is there a better term for "non-negative"?

Real numbers $x$ satisfying $x \geq 0$ are said to be non-negative. Some alternative phrases include positive or zero and at least zero. However, I find these phrases to be unsatisfactory, for a ...
• 65.1k
1 vote
46 views

### an ordered abelian group has no order units

An element $e$ in $G^{+}$ is called an ordered unit in an ordered abelian group $(G,G^{+})$ if for any $g\in G$,there exits a positive integer such that $-ne\leq g \leq ne$. In Rordam's book,there ...
• 3,013
44 views

### isomorphism in ordered monoids

I read that a morphism $\gamma : S \rightarrow T$ is an isomorphism if there exists a morphism $\Psi : T \rightarrow S$ such that $\gamma \circ \Psi = I(T)$ and $\Psi \circ \gamma = I(S)$, where $I$ ...
• 705
114 views

### Order preserving injection $f$ from set of rationals $Q$ into $R$ with discrete image.

How to construct an order-preserving injection $f:Q\rightarrow R$ , such that the image of $f$ is discrete subspace of $R$ (set of reals).
• 1,190
42 views

### Looking at Automorphisms of Subgroups of $(\mathbb R,+)$ With Positive Slope

Let $(R,+)$ be a non-trivial subgroup of $(\mathbb R,+)$. We say that an automorphism $\phi$ of $R$ has positive slope if $\tag 1 \phi(R \cap \mathbb R^{\gt 0}) \subset \mathbb R^{\gt 0}$ What ...
• 10.5k
1 vote
65 views

### Continuity of absolute value in topological ordered abelian groups

Let $(G,+,0)$ be an abelian topological ordered group, that is, $G$ is endowed with a total order $\leq$ such that, for any $a,b,c\in G$, we have that $a\leq b$ implies $a+c\leq b+c$. Moreover, $G$ is ...
• 5,901
1 vote
213 views

### Totally ordered abelian group with a unique " isolated subgroup"

Let $(G,+,<)$ be a totally ordered abelian group i.e. $(G,+)$ is an abelian group with partial order $<$ such that for every $a,b\in G$, exactly one of $a=b$ or $a<b$ or $b<a$ holds; and ...
• 3,583
143 views

### Ordered groups: do irreducible elements always commute?

The title is the main question. Explaining the notation, an ordered group is a group $X$ with a partial order such that for every $x,y,z\in X$, if $x\leq y$, then $zx\leq zy$ and $xz\leq yz$. An ...
• 883
102 views

### Definition of totally ordered monoid?

Suppose I have the monoid $(\mathbb{N},\times)$. It is my understanding that for the relation $\le$ to form a total order on the above monoid, the following must be true: a\le b\iff (\forall c\in\...
1 vote
93 views

### On totally ordered abelian groups having exactly one convex subgroup

Let $(G, <)$ be a totally ordered abelian group. Let us call a proper subgroup $H$ of $G$ to be convex if for every $a \in H$, $[a,-a]:=\{x\in G : -a \le x \le a\} \subseteq H$. If $G$ has exactly ...
• 4,224
122 views

### Does every ordered divisible abelian group admit an expansion (and how many) to an ordered field?

Let $(G,+,<)$ be an ordered divisible abelian group. $1)$ Is it always the case that there exists a binary function $*:G\times G \rightarrow G$ such that $(G,+,*,<)$ is an ordered field? $2)$ ...
• 2,881
38 views

### Uniqueness of total orders on abelian groups

Suppose $G$ is an abelian group totally bi-ordered by $\leq$ and by $\leq'$. Does it follow that $\leq'$ is either equal to or the converse of $\leq$?
• 509
103 views

### On the definition of the totally ordered abelian group $\Gamma/\Delta$

If $\Gamma$ is a totally ordered abelian group, and $\Delta\subset\Gamma$ is a convex subgroup (meaning if $\delta,\delta'\in\Delta$ and $\delta\leq\gamma\leq\delta'$ then $\gamma\in\Delta$), then we ...
56 views

### Suppose a ring R is a PID. What does this say about possible orders on the set of R?

More explicitly, is there always some partial or total order we can apply to the set of R?
35 views

• 2,881
51 views

### If ordered ring $A$ has upper or lower bound, then, $A = \{0\}$ (alternative proof)

Knowing that any ordered ring $A \neq \{0\}$ is infinite, can I say that if an ordered ring is finite, it must be $\{0\}$ ? By this result - If an ordered ring $A$ has an upper bound, does it have ...
1 vote
79 views

### Counting permutations without fixpoint

Let $n$ be a positive integer. Consider an ordered set $S_n = [1,2,3,...,n]$ where the $j$ th element from the left equals $j$. Now consider a function defined on $S_n$ as a permutation of that set. ...
• 12.9k
95 views

### Theory with non archimedean models

Is there a consistent theory in the language of ordered groups (or ordered rings) whose models are non archimedean ordered groups (or rings or fields)? (note: I am not asking for the existence of an ...
• 4,739
1 vote