All Questions
Tagged with well-orders abstract-algebra
10
questions
0
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1
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56
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Reordering algorithm to fragment consecutive sequences of ones as much as possible
Recently, I came across the following problem:
Let $s_1, s_2, ..., s_k$ be non-empty strings in $\{0,1\}^*$.
We define $S_{s_1,s_2,...,s_k}$ as the concatenation of $s_1, s_2, \dots, s_k$.
We call a &...
- 1
1
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0
answers
81
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Every Non empty subset has a least element implies linear order
Suppose $(A,R)$ be structure where R is a binary relation on $A$.
Suppose $A$ has the property that every Non empty subset of $A$ has a least element w.r.t. the relation $R$. Then $R$ is a linear ...
user1072974
1
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0
answers
85
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State if each of the objects' set is well-defined or not.
Q. 1.1 taken from book titled: First-Semester Abstract Algebra:
A Structural Approach, by: Jessica K. Sklar.
State if each of the below objects stated below is a well-defined set or not.
$\{z\in \...
- 4,532
2
votes
2
answers
50
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Does sign-magnitude ordering admit infinite descending chains in any discretely ordered ring?
I have a conjecture:
Let $R$ be an ordered ring $(S, 0, 1, +, ·, <)$.
Let $|x| = x$ if $0 \leq x$ and $|x| = -x$ if $x < 0$.
Define $x \preccurlyeq y$ to hold if either $|x| < |y|$ or both $|...
- 688
1
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2
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67
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Given three equivalent statements, prove equality between two sets of variables
What technique should I use to solve the following problem?
Would I utilize the division algorithm?
Let $ m, n, r, s ∈ \mathbf{Z}$.
If $m^2 + n^2 = r^2 + s^2 = mr + ns$,
prove that $m = r$ and $n = s$....
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93
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Showing that in a well ordered ring homomorphic to $\mathbb{Z}$, there are no elements between zero and the ring's identity. [duplicate]
I'd like to prove the following statement:
Given a well-ordered ring $A\neq \emptyset$ with identity $I$ and a
homomorphism $\phi: \mathbb{Z}\to A$ such that $\phi(n) = nI$, then
the set
$...
- 315
-1
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1
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76
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assumption of well-ordering principle
An abstract algebra book says "the basic assumption here is 'well-ordering principle' : any nonempty set of nonnegative integers has a smallest member." and then using it, the book proves mathematical ...
user682705
1
vote
1
answer
36
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Well ordering of $\mathbb{N}$ with weak induction
In my Algebra course the well-ordering property of $\mathbb{N}$ has been proven just by standard ("weak") induction, but I can't understand the proof given by the lecturer.
The proof proceeds as ...
- 579
2
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1
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77
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A proof in module theory: a set theoretic consideration $|\bigcup_{i \in I} B_i| \leq |Y|$
The set-theoretic result used in the proof I want to know about is the following:
Let $(B_i)_{i \in I}$ be a family of sets such that, for all $i,j \in I$, either $B_i \subseteq B_j$ or $B_j \...
- 4,528
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Prove that every finitely generated subgroup of a totally ordered group can be totally ordered.
Prove that every finitely generated subgroup of a totally ordered group can be totally ordered. A group can be totally ordered if there is a total order $\leq $ on that group.
Is the proof of this ...
user482939