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Questions tagged [ordered-groups]

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

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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 $I$ ...
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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).
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How to structure a written generic expression for an ordered pair

My teacher gave the following exercise "For each of the following definitions, give a graph of the function. Say whether this is a partial or a total function on real numbers. If the function is ...
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Permutation of ordered pairs

$\DeclareMathOperator*{\maxi}{maximize}$ Let $Z$ be a set of n ordered pairs, defined as $Z = \{(a_{k}, b_{k}) | a_{k}, b_{k} \in \mathbb{R}, k \in [1, n]\}$. We define a permutation $\pi$ of the ...
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Homomorphism of groups, subgroups

If $f: G_1\to G_2$ is homomorphism of groups $G_1$ and $G_2$ and if $|G_2|=25$ and $A$ is subgroup of $G_1$ such that $A\neq \{e\}$, and $f(A)\neq G_2$, prove that $f(A)$ is a subgroup of $G_2$, and ...
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25 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 ...
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Index to Value for a Set of Tuples, Ordered by the Sum of a Tuple's Values.

Order the set of all $n$-tuples by their sum, $\sum$, and an index, $i$. Formulaically, find a particular tuple given only these three numbers. The following table shows the first few tuples '...
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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 ...
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Partially ordered? Transitive closure?

We are given R = {(1, 1), (2, 1), (2, 2), (2, 4), (3, 1), (3, 3), (3, 4), (4, 1), (4, 4)}. It is reflexive because (1,1), (2,2), (3,3), (4,4) is in the set. It is antisymmetric because (b,a) is not ...
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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 ...
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Is every pogroup generated by its own prime elements isomorphic (as pogroup) to some $\mathbb{Z}^{(I)}$?

A pogroup (partially ordered group) is a group with a partial order $\leq$ such that if $x\leq y$ then $zx\leq zy$ and $xz\leq yz$. A prime element of a pogroup is an element $x$ such that $x>e$ ($...
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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 ...
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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\...
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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 ...
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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? ...
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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 $\leq$?
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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 ...
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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?
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a property of a morphism of ordered groups

An ordered group $(G,G^+)$ is an abelian group $G$ together with a subsemigroup $G^+$ containing the identity $0$, having these properties: $G^+-G^+=G$. $G^+\cap (-G^+)=\{0\}$. We have for $x,y\in ...
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Subgroups of the rationals is either generated by one element or infinitely generated

The following situation is given: Consider $(\mathbb{Q},+)$ as an ordered abelian group: $x,y\in \mathbb{Q}, \; x\le y\; :\Leftrightarrow y-x\ge 0$. Let $\mathbb{Q}_+=\{c\in\mathbb{Q}\mid c\ge 0\}.$ ...
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When does an abelian “linearly ordered group ” has the property that any non-empty subset of the set of “non-negative” elements has a “least” element

By a theorem of F.W. Levi , we know that an abelian group can be equipped with a linear order (https://en.wikipedia.org/wiki/Linearly_ordered_group) iff the group is torsion free . So let $(G,\le )$ ...
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Existence of a certain near-metric map on an ordered divisible abelian group

Let $\mathcal{M}=(M,0,+,<)$ be a linearly ordered divisible abelian group. Let's define an $\mathcal{M}$-metric on $M$ to be a map $d:M\times M \rightarrow M$ such that (1) $\forall x,y\in M,\, d(...
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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 ...
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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. ...
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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 ...
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Density definition in totally ordered Abelian group

In my analysis textbook, there's a section about totally ordered Abelian groups, and there's a definition of density, which goes as follows: A subset $A \subset G$ of a totally ordered Abelian group $...
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123 views

Well-ordered sets in the field of Hahn series

Given a linear order $(E,<)$, let $o(E)$ denote the least ordinal which does not order-embed in $(E,<)$. It is known that for an ordered field $k$, $o(k)$ is a regular ordinal $\geq \omega_1$. ...
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Existence of a sequence converging to $0$

Let $(G,+,\le)$ be a partially ordered group (with identity $0$) and suppose that for each positive $g \in G$ there exists $g^\prime \in G$ such that $0<g^\prime<g$. Is it true that there ...
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Defining a subsegment from an ordered set using mathematical notation

I have a ordered set of time stamps events $T$ indexed by $t$;I also have a set of observations at each time stamps. $O_{T}$ is my set of observations indexed by $O_t$ where $O_t$ is a observation at $...
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Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$?

Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$? (With lexicographical order) Thanks
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Ordered ring and mutiplicative ordinal

If $(E,<)$ is a linear order, let $s(E)$ denote the supremum of the set of ordinals which (order-)embed in $(E,<)$. $s(E)$ is also the set of ordinals which embed in $E$ with a non cofinal range....
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Probability to find at least one alphabetically ordered subset of K elements in a set of N elements

I would like to know how to calculate the probability of finding an alphabetically ordered subset of at least K elements in a set of N alphabetical elements. For example, for a set of N letters from ...
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804 views

Intervals in divisible ordered groups

Is it true that if $(G,+,0,<)$ is a divisible ordered abelian group with at least two elements, then for $a,b >0 \in G$, there is an injective order preserving map from $[0;a)$ to $[0;b)$? It ...
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On ordered abelian groups containing $\mathbb{Z}$

Let $\Delta$ be an ordered abelian group containing $\mathbb{Z}$ as a subgroup of index $e$. I need to show that for any positive element $\delta \in \Delta$, we have $e\delta \geq 1$. I have no ...
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Can the order on an ordered, cancellative monoid be extended to its Grothendieck group?

Suppose we have an ordered, cancellative monoid and we wish to apply the Grothendieck group construction to it. Can the total order be extended to the larger group? Example: consider the ordered ...
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Partial order relation on the group of integers

We know that the usual $\leq$ is a partial order relation on the group of integers $\mathbb Z$ and $\mathbb Z$ is a totally ordered with this partial order relation. Is there any other partially order ...
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Equivalent condition of spliting exact sequence of partially ordered groups

A short exact sequence $ 0 \rightarrow A \rightarrow B \rightarrow C\rightarrow 0$ of partially ordered group, where $\alpha : A\rightarrow B$ and $\beta: B\rightarrow C$ are order homomorphism is ...
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Abstract Algebra: Every group has a cyclic subgroup

I have to show that every group has a cyclic subgroup. I know what this means, and to me it is obvious, yet I am not sure how to formally write it. I proved it directly, as follows: Let G be a ...
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An $RO$-group which is not $O$-group

I was thinking of some example for an Right ordered group ( $RO$-group) which is not an $O-$group (Ordered group) i.e. not left ordered. I guess looking in matrix groups will be fruitful but how to ...
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2answers
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Can every torsion-free nilpotent group be ordered?

I know that a torsion-free abelian group can be ordered and have done two proofs for that too. But the next two question that popped up in my mind were- Can every torsion-free nilpotent group be ...
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Atoms of the Lattice of $\ell$-ideals of a lattice ordered group

I'm new to lattice groups and I'm stuck in the very first proposition (2.1) of Paul Conrad's paper "Characteristic Subgroups of Lattice-Ordered Groups" (http://www.jstor.org/stable/1995910). Part (c) ...
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Left orderable Group has infinite order

An $order$ on a set $S$ is an (anti-symmetric) relation $<$ on $G$ so that for each $a,b\in G$ exactly one of the following is true: $a<b, b<a$ or $a=b$. A group $G$ is called left orderable ...
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Free ordered vector space over an ordered abelian group

Let $G$ be a partially ordered abelian group (written additively). I want to add $\mathbb{R}$-multiples to $G$ in a "free" way ,thus extending $G$ to an ordered vector space. Construction: To this ...
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An ordered group $G$ is Archimedean if and only if the following holds…

Let $G$ be an ordered group; then $G$ is Archimedean if and only if the following condition holds: $$\text{if} \space a, b \in G \space \text{with} \space a>0, \space \text{ there exists a natural ...
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Creating Bratteli diagrams for Riesz groups

The Effros-Handelman-Shen-theorem tells you that Riesz groups are the same as dimension groups -- i.e. any ordered, unperforated abelian group with the Riesz interpolation property can be realised as ...
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630 views

Order Preserving Isomorphism

If abelian group $G$ has an archimedean order then there is an order preserving isomorphism $\phi$ of $G$ onto a subgroup of $\mathbb{R}$. Here we can say that $G$ is archimedean totally ordered ...
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Can all non-archimedean groups be written as a product of archimedean groups?

We say that a partially ordered group $(G,\cdot, \geq)$ is Archimedean if for any $g,h >1\in G$ there exists some n such that $g^n > h$. All the non-archimedean groups I know of can be written ...
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Every torsion free divisible abelian group admits an order compatible with the group operation

I am looking for a proof of this result. I found a proof which goes like this: Call the group $G$. Take a maximally independent {$r_a$} and totally order it. Every element of $G$ can be written as a ...