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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
revised Which associative and commutative operations are defined for any commutative ring?
edited body
Nov
1
revised Which associative and commutative operations are defined for any commutative ring?
fixed sign mistake and updated question with information from the comments
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.)
Nov
1
asked Which associative and commutative operations are defined for any commutative ring?
Oct
31
answered Can the b-adic representation of rational numbers (by quote notation) be extended to non-terminating expansions?
Oct
30
reviewed Approve How to prove this inequality $\frac{a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n-1}a_{n}}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}\le\cos{\frac{\pi}{n+1}}$
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
30
asked Can the b-adic representation of rational numbers (by quote notation) be extended to non-terminating expansions?
Oct
28
awarded  Revival
Oct
16
reviewed Approve coordinate geometry high level problems
Oct
7
reviewed Approve infinitely many solutions to $\displaystyle x^n + y^n = z^{n+1}$
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.
Oct
6
revised Are higher order logics substantially stronger than second order
Oh, I forgot that I wanted to say something about the last word property
Oct
6
answered Are higher order logics substantially stronger than second order
Oct
5
revised Is multiplication in mixed radix numeral systems complicated?
edited tags
Oct
4
asked Is multiplication in mixed radix numeral systems complicated?