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Apr
13
comment Does the “equality semigroup” have an accepted name?
If you introduce the name "subatom" for an element which is either the bottom or an "atom", then you can no longer pretend that you are using established terminology. If you slightly abuse existing terminology on the other hand, then you are just following established mathematical practice. (red herring principle...)
Apr
13
comment Does the “equality semigroup” have an accepted name?
In any partial ordered set with a bottom element, the meet-semilattice of atoms (en.wikipedia.org/wiki/Atom_(order_theory)) is a well-defined semilattice. So I would call it just this: "meet-semilattice of atoms".
Mar
17
comment Is there a projective metric on a projective space induced by a p-norm?
cross posted at mathoverflow.net/questions/199839/…
Feb
21
comment Can every rational function be represented in barycentric form?
@JohnHughes You mean because $y_j$ is not just a coefficient, but also identical to $r(x_j)$? Maybe you have a good point, and it could be related to my troubles understanding the cited statement.
Feb
21
comment Can every rational function be represented in barycentric form?
In Barycentric Lagrange Interpolation. Jean-Paul Berrut, Lloyd N. Trefethen., the earliest references seem to be from 1997 of Jean-Paul Berrut. This is a case of an author citing another paper of himself as reference, so "it is known" might only refer to a very small set of people...
Dec
23
comment What's behind the Banach-Tarski paradox?
The physical world doesn't deal with sets at all!
Nov
29
comment Which associative and commutative operations are defined for any commutative ring?
@ThomasAndrews Yes, in case of addition the corresponding condition is that $b$ has to be in the range of $x\to ax$. What I like less is that $a$ and $b$ are currently not true parameters, but are restricted to the free commutative ring without generators, i.e. $\mathbb Z$. But I'm not sure how to extend Martin Brandenburg's "natural" argument to the case with parameters.
Nov
29
comment Which associative and commutative operations are defined for any commutative ring?
@ThomasAndrews It certainly won't be in the range of $\phi_{ab}$ if $b(b-1)/a$ is not defined, i.e. when $b(b-1)$ is not in the range of $x\to ax$. If on the other hand $b(b-1)$ is in the range of $x\to ax$, then the laws of commutative rings are sufficient to show that also the other expression will work.
Nov
29
comment How to show distributivity in a ring, and what is wrong with my algebra?
See also the appendix of math.stackexchange.com/questions/1001186/…
Nov
6
comment Graph-Minor Theorem for Directed Graphs?
@Denis My guess would be that the other answer was the first answer by a new user, and got its votes through the corresponding review queue. In the review queue, you can only see the answer and the question, but not the other answers. The main claim I make in this answer is that the definition "A directed graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges." is equivalent to the definition given in the question. If you want to challenge this claim, or think I should it explain it in more detail, please let me know.
Nov
3
comment Example of a commutative ring which is not a subring of a commutative ring where every non-invertible element is a zero-divisor
Thanks for the link. So instead of "every non-invertible element is a zero-divisor", I could say "every regular element is a unit". Sweet, short and positive...
Nov
2
comment Which associative and commutative operations are defined for any commutative ring?
Wow, you really taught me some category theory today. The fact that ring homomorphisms don't preserve non-units is so typical, yet I still fall into the "law of excluded middle" trap. And the lesson about natural transformations and representation of the forgetful functor was most welcome, since "theoretically I mastered that material some time ago", but this "practical application" was still challenging for me. It took me more than an hour and some repetition of material which I "believed to master", before I got the warm fuzzy feeling that I understood what you told me...
Nov
1
comment Which associative and commutative operations are defined for any commutative ring?
Maybe the "if $x\in R^*$ then $f(x)$ else $g(x)$" can't be interpreted as a natural transformation, but before we can even ask whether an operation is a natural transformation, we have to fix the source and target functor. This was the crucial point why it took me so long to come to terms with adjoint functors: I kept thinking about natural transformations without taking care that the source and target functor are as crucial as the family of morphisms itself. Your diagram looks as if the target functor is an arbitrary $\mathcal F$, and the source is $\mathcal F \times \mathcal F$. Correct?
Nov
1
comment Which associative and commutative operations are defined for any commutative ring?
It should be possible to define operations in rings with the help of an "if $x\in R^*$ then $f(x)$ else $g(x)$" construct, where $R^*$ is the group of units of $R$. You argued that this construct is not natural in the sense of category theory, but I fail to see why it should not be natural, as long as $f(x)$ and $g(x)$ are natural. The property of being an element of the group of units seems to be invariant under (unitary) ring homomorphism to me.
Nov
1
comment Which associative and commutative operations are defined for any commutative ring?
@MartinBrandenburg I think I already found a counterexample to my question. Take the unary operation $u(x)$ which maps any element of the group of units to $1$, and all other elements to $0$. Then $u(xy)$ is an associative and commutative binary operation not covered by $x_{abc}y$. By "operation", I mean that only the ring structure is used, and not an order or field structure, which wouldn't be available in a general ring. But I don't put any special requirements on $a, b, c$, like being simultaneously defined for all rings. (I downloaded you reference and will read it now.)
Oct
30
comment Can the b-adic representation of rational numbers (by quote notation) be extended to non-terminating expansions?
I also thought about this. My only worry was that the $p$-adic metric is induced by an absolute value (i.e we have $|xy|_p=|x|_p|y|_p$), but the combined norm only satisfies $|xy|\leq|x||y|$. So I wonder whether this is enough to ensure that the completion is a field.
Oct
30
comment Can the b-adic representation of rational numbers (by quote notation) be extended to non-terminating expansions?
I see a potential problem now, if I only check whether the arithmetic operations are well defined: After the addition of two numbers in the representation, one of the $\beta_i$ might go towards infinity. Hence the result must be set to zero. Which is fine in a certain sense, but associativity (and distributivity) of addition might be lost (not sure whether this can happen). So in addition to being well defined, I also have to check that addition is associative and distributive.
Oct
7
comment Are higher order logics substantially stronger than second order
In a certain sense, this answer is only true if the other axioms ensure that there are infinitely many objects/elements in the universe. And if the other axioms already ensure that we have the consistency strength of bounded Zermelo set theory or ZFC set theory, then just adding higher order variables and impredicative comprehension axioms (without using higher order variables in some of the other axioms) won't increase consistency strength any further.
Sep
28
comment Why are real numbers useful?
Nice answer, but when you say "And the wanted this line to be with no gaps, to be a continuum," what do you mean by "no gaps"? The rational numbers have no gaps either, in a certain sense. In another sense, the real numbers have gaps too, for example if you look at them as a subset of the surreal numbers. I think instead of the unclear "no gaps", what they really wanted was "completeness" (not necessarily limited to the order) and the Archimedean property: $\forall x\in {\mathbb R} \quad \exists n\in {\mathbb N} \quad x < n$.
Jul
22
comment Why are box topology and product topology different on infinite products of topological spaces?
@MathsLover The set theoretical topological spaces where defined by Felix Hausdorff in his book "Grundzüge der Mengenlehre", which appeared in 1914. In 1912, Jan Brouwer had started intuitionism, but Felix Hausdorff's work is not really based on it. The linked publication page of Dirk van Dalen is a good source for the connections between formal intuitionistic logic and topological spaces. If the links in that page don't work in your browser, copy the desired "link address" and replace "papers.html" in the current addresss by "articles/..." from the copied "link address".