# Questions tagged [ordered-groups]

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

48 questions
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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 $I$ ...
1answer
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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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### 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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