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$$x\neq y\in S$$

## Infinite Hausdorff space that is not homeomorphic to any proper quotient

https://mathoverflow.net/questions/200613/infinite-hausdorff-space-that-is-not-homeomorphic-to-any-proper-quotient
Let S be a set and \vartheta be an equivalence relation on S . We say that \vartheta is proper if there are x\neq y\in S with (x,y)\in\vartheta . Is there an infinite Hausdorff space (X,\tau) such that for every proper equivalence relation \vartheta on X we have X\not\cong X/\vartheta ?...
$$x\neq y \in \kappa$$

## What is a hypergraph minor?

In a note concerning a version of Hadwiger's conjecture for hypergraphs, I once defined what a (complete) minor map of a hypergraph is, and probably this can be used to define a notion of "minor of hypergraph". (If you are interested in the note, which I did not publish or put on arXiv, I am happy to send it to you.) For the following definitions assume that H=(V,E) is a hypergraph. A subset \emptyset \neq S \subseteq V is said to b...
$$x^p = y \in S$$

## $k[[x]]$ as a $(k[[x]])^p$ module for ugly fields

The answer to the question is no. Let R = k[[x]] , S = k[[y]] , and f:R \to S is the absolute Frobenius. We must show that S is not a free R -module. By assumption on k , we know that S must have infinite rank if it were free. On the other hand, S is x -adically complete (since x^p = y \in S ). Hence, the claim follows from: Lemma: The R -module M := R^{\oplus I} is x -adically complete if and only if I is finite. Pr...
$$E(G(k,S)) = \big\{\{x,y\}: x\neq y \in S \text{ and}y\in S_k(x) \text{ and }x\in S_k(y)\big\}.$$

## Graphs represented by a subset of a metric space

https://mathoverflow.net/questions/313047/graphs-represented-by-a-subset-of-a-metric-space
Let (X,d) be a metric space, and suppose S\subseteq X is a finite subset in which all pairwise distances are distinct (formal definition here). If x\in S and k is a non-negative integer with k<|S| , we define S_k(x)\subseteq S to be the set of the k nearest members of S to x , other than x itself. We define a graph G(k,S) by setting V(G(k,S)) = S , and E(G(k,S)) = \big\{\{x,y\}: x\neq y \in S \text{ and } y\in S_k...
$$d^H_{n+1}(f(x), f(y)) = d^H_n(x,y) + 1 \text{ for all}x\neq y\in S$$

## Stretching map of $n$ points from $\{0,1\}^n$ to $\{0,1\}^{n+1}$ with respect to their Hamming distance

https://mathoverflow.net/questions/319131/stretching-map-of-%24n%24-points-from-%24%5C%7B0%2C1%5C%7D%5En%24-to-%24%5C%7B0%2C1%5C%7D%5E%7Bn%2B1%7D%24-with-respect-to-their-hamming-distance
Given a positive integer n , the Hamming distance d^H_n(x,y) of x,y\in \{0,1\}^n is defined by d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|. Given an integer n>0 and a set S\subseteq \{0,1\}^n with |S| = n , is it possible to find a map f:S\to \{0,1\}^{n+1} such that d^H_{n+1}(f(x), f(y)) = d^H_n(x,y) + 1 \text{ for all } x\neq y\in S ?...
$$v=xy \in S_{\alpha}$$

## Characterizing subfields $\mathbb{C}(u,v) \subseteq \mathbb{C}(x,y)$ invariant under an involution

https://mathoverflow.net/questions/354155/characterizing-subfields-%24%5Cmathbb%7Bc%7D%28u%2Cv%29-%5Csubseteq-%5Cmathbb%7Bc%7D%28x%2Cy%29%24-invariant-under-an-involution
Let \iota be an involution on \mathbb{C}(x,y) , namely, a \mathbb{C} -algebra automorphism of \mathbb{C}(x,y) of order two. Examples of involutions: \alpha: (x,y) \mapsto (y,x) , \beta: (x,y) \mapsto (x,-y) , \epsilon: (x,y) \mapsto (-x,-y) . Observe that \alpha and \beta are conjugate, while \epsilon is not conjugate to them. (The Jacobian of \alpha and \beta is -1 , while the Jacobian of \epsilon is 1 ). Assume ...
$$x,y\in S.$$

## Cancellable elements of a power semigroup

https://mathoverflow.net/questions/192157/cancellable-elements-of-a-power-semigroup
For a semigroup S, its power semigroup P(S) is the semigroup of all non-empty subsets of S with the operation given by AB=\{ab\,|\,a\in A,b\in B\}. I would like to know about the cancellable elements of P(S) given some knowledge of cancellability in S. If s\in S is left-cancellable in S, then \{s\} is also left-cancellable in P(S) because if sA=sB and a\in A, then sa=sb for some b\in B, and a=b\in B follows...
$$y\lt x\in S\to y\in S$$

## A principle of mathematical induction for partially ordered sets with infima?

Something very close to François' conditions achieves the desired if-and-only-if version of the theorem for partial orders, providing an induction-like characterization of the complete partial orders, just as Pete's theorem characterizes the complete total orders. Suppose that (P,\lt) is a partial order. We say that it is complete if every subset A has a least upper bound sup(A) and a greatest lower bound inf(A) . This implies th...
$$xy \in S.$$

## Arithmetic closed subsets

https://mathoverflow.net/questions/67618/arithmetic-closed-subsets
Let S, S_1 be subsets of the positive numbers \mathbb{N} . We say (as usual) that S is multiplicative closed if x \in S and y \in S implies xy \in S. We say also that S_1 is arithmetic closed if x \in S_1 and y \in S_1, and \gcd(x,y)=1, implies xy \in S_1. Let T be a subset of the positive integers containing 1 and at least another element. The smallest multiplicative closed set that contains T (say C(T) ) i...
$$x,y\in S$$

## Idempotents in Green J classes

Any regular J -class with a joined zero is a 0-simple semigroup. So your question is reduced to the following: who many idempotents has a 0-simple semigroup? In particular, if S is finite, a J -class (with 0) is completely 0-simple semigroup, so it has just one idempotent \ne 0 iff it is a group. Moreover, if a 0-simple semigroup S with 1 has no other idempotents, then it is a group with 0. Proof: Let G be its subgroup of inve...
$$x,y\in S$$

## An extremal property of points on the unit sphere of a 2-dimensional Banach space

https://mathoverflow.net/questions/344612/an-extremal-property-of-points-on-the-unit-sphere-of-a-2-dimensional-banach-space
Let (X,\|\cdot\|) be a 2-dimensional real Banach space and S=\{x\in X:\|x\|=1\} be its unit sphere. Assume that S is smooth in the sense that for any y\in S there exists a unique functional y^*:X\to\mathbb R such that y^*(y)=1=\|y^*\| . This unique functional y^* will be called the supporting functional at y . Let x,y\in S be points such that \|y-x\|+\|y+x\|=\max\{\|s-x\|+\|s+x\|:s\in S\}. Question. Is y^*(x)=0 ?...
$$x,y \in S$$

## Example that a finite collection of contractions can't approach the set of all contractions well enough

https://mathoverflow.net/questions/224237/example-that-a-finite-collection-of-contractions-can%27t-approach-the-set-of-all-contractions-well-enough
I'm looking for an example of a seperable and complete metric space (S,d) such that there exist some \varepsilon > 0 and P a probability measure on the Borel open sets \mathcal{B}_S for which \int d(x,a) P(dx) < \infty for some a \in S such that for any finite collection F of non-expansive maps from S to \mathbb{R} , there exists some probability measure Q on B_S for which there exists some a \in S: \int d(x,a) P(d...
$$x,y \in S$$

## Li-Yorke chaos: the non compact case

https://mathoverflow.net/questions/299083/li-yorke-chaos%3A-the-non-compact-case
1) Is there any notion of Li-Yorke chaos for non compact (metric) spaces X and non continuous transformation f:X \rightarrow X ? Could you bring some references? 2) I mean, why are so important the compactness of the space X and continuity of the function f:X \rightarrow X in the Li-Yorke chaos definition? Just to remember, a pair x,y \in X is called scrabled if \liminf d(f^{n}(x),f^{n}(y)) = 0 and \limsup d(f^{n}(x),f^{n}(y...
$$x,y\in S,\ x\ne y\implies x-y\notin\mathbb Q$$

## Picking a real for every non-empty open set in $\mathbb{R}$

With the axiom of choice: if \kappa is an infinite cardinal, then any collection of (at most) \kappa sets, each of cardinality (at least) \kappa , has an injective choice function. In this case \kappa=2^{\aleph_0} , and not only the collection of all nonempty open subsets of \mathbb R but even the collection of all uncountable Borel subsets of \mathbb R has an injective choice function. Without the axiom of choice: Construct a...
$$\forall x,y\in S: x\ne y\implies h(x,y)\ge k$$

## Binary codes with upper bound on pairwise distance

https://mathoverflow.net/questions/141562/binary-codes-with-upper-bound-on-pairwise-distance
A fundamental problem in coding theory is: Given positive integers k\le n , determine or bound the maximal cardinality of a set S\subset \mathbb{F}_{2}^{n} such that \forall x,y\in S: x\ne y\implies h(x,y)\ge k , and construct such sets. (Here h is Hamming distance.) What about the complementary idea? Replace h(x,y)\ge k with h(x,y)\le k . I'm sure these have been explored before, so keywords and references would be appreciated...
$$\forall x,y\in S: x\ne y\implies |\langle x,y\rangle|\le\epsilon$$

## Almost orthogonal vectors in subsets of euclidean space

https://mathoverflow.net/questions/131554/almost-orthogonal-vectors-in-subsets-of-euclidean-space
Given the vector space \mathbb{R}^n , endowed with the standard inner (dot) product \langle\cdot,\cdot\rangle:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R} , the problem of almost-orthogonal sets asks, for each \epsilon> 0 , for the construction (or at the very least, bounds on the cardinality) of a maximal subset S of U_n , the set of unit vectors of \mathbb{R}^n , such that \forall x,y\in S: x\ne y\implies |\langle x,y\rangle|\...
$$y\in S$$

## Can we classify all finite 2-generated groups $G$ such that if $x,y$ generate $G$, then so does $x,yxy^{-1}$?

Theorem The only finite group satisfying the condition is the trivial group. Proof. Let G be a nontrivial finite group and S be a simple quotient of G , which satisfies the condition by Gaschutz's lemma. Then S is non-abelian as mentioned above. If x\in S is any involution, then by well-known result of Guralnick and Kantor in Probalistic generation of finite simple groups, there exists an element y\in S such that S=\langle x...
$$y\in S^{N-1}$$

## improved Sobolev embedding

https://mathoverflow.net/questions/368241/improved-sobolev-embedding
This is probably not a research level question but I am struggling with the geometry. My question is related to whether some monotonicity can increase the range of exponents in the Sobolev embedding. For instance on the unit ball, nonnegative radially symmetric functions which are nondecreaing in the radial direction should satisfy a Sobolev embedding for an improved range of exponents: Indeed such functions must be large on the full bo...
$$\forall x, y \in S: x \neq y \implies (x \rhd y = y \iff y \rhd x \neq x).$$

## Shelves with "trichotomy"

https://mathoverflow.net/questions/328269/shelves-with-%22trichotomy%22
A left shelf (S, \rhd) is a magma with the left self-distributive law: \forall x, y, z \in S: x \rhd (y \rhd z) = (x \rhd y) \rhd (x \rhd z). Shelves are generalization of racks and quandles from the knot theory. I am looking for examples of shelves with the following additional axiom: \forall x, y \in S: x \neq y \implies (x \rhd y = y \iff y \rhd x \neq x). Can a left shelf satisfy this property? (See also the follow-up ...
$$x\in R\in C,y\in S\in C$$