Questions tagged [ordered-groups]
An ordered group is a group with a (partial) order which the group operation preserves.
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Is an Archimedean topological group of the reals isomorphic to $(\mathbb R,+)$?
A group $H:=(\mathbb R,\boxplus)$, is given to be
Ordered as per the canonical order of $\mathbb R$.
Archimedean as per order in 1.
Topological as per canonical topology of $\mathbb R$.
Can it be ...
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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 ...
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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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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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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 ...
user472520
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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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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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The composite valuation
I am currently studying valuation theory and came across the concept of composite valuation. I think, I proved that the composite valuation is always given by a lexicographic product of the two ...
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Question about ordered group and its order ideal.
I am reading a book entitled An introduction to the classification of amenable C*-algebras.
It reads,
Definition 3.3.1 An ordered group $(G,G_+)$ is an abelian group $G$ with a distinguished ...
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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 ...
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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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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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Characterization of weighted total orders on $\mathbb{N}^r$ or $\mathbb{Z}^r$
By a weighted total order on $\mathbb{N}^r$ or $\mathbb{Z}^r$ I mean the relation defined by
$$
(m_1,\ldots,m_r) \leq (n_1,\ldots,n_r)
\quad\Longleftrightarrow\quad
\sum_{i=1}^r c_i m_i \leq \sum_{i=1}...
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if f is a homomorphism from L to L', Should the image f(L) be a sublattice of L'?
I'm a beginner in the subject & my question can be meaningless, so I'm sorry from start if that's the case.
I just don't understand why all of the image f(L) can be a sublattice of L' when f is a ...
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