Commutative Monoid and some properties

  • Thread starter Kindayr
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  • #26
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What if we work in [itex]\mathbb{Z}^{n}_{p}[/itex] over a the field [itex]\mathbb{Z}_{p}[/itex], as I discuss in the second part of my paper? This gets a bit messy I guess though.
Aaah, that might work :smile:
 
  • #27
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Aaah, that might work :smile:
Then I will play around with that tonight, and type some stuff up tomorrow! I'll try to generalize everything using the map you defined, and see what I can do.

thank you very much micromass!
 
  • #28
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I'm just trying to find a way to work this.

Fix [itex]p[/itex] prime. Let [itex]1<p_{1}<p_{2}<\cdots<p_{n}<p[/itex] be all primes between [itex]1[/itex] and [itex]p[/itex], and let [itex]\mathbb{Z}_{p}^{n}[/itex] be the set of all [itex]n[/itex]-tuples with entires in [itex]\mathbb{Z}_{p}[/itex]. Then our map [itex]\Phi[/itex] doesn't work since we can find a [itex]q\in\mathbb{Q}^{*}[/itex] whose factorization uses a prime [itex]p_{0}>p[/itex].

So that's just where I'm confused.

I'm just trying to get over a slump on how to turn this into a vector space.
 
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  • #29
Hurkyl
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Whoa, could you please explain? I haven't been exposed to much of any metamathematics.
Once you have a data structure, you can do a lot. Just look at set theory. :smile:

Similar power is opened up to you once you can view an integer as a list of integers.

There are a variety of related things, I'll go down the route of computability -- what is a computer? It is essentially just it's current state -- a list of integers (e.g. the list might contain 16 registers, and the rest be memory which can be interpreted as instructions or data) -- and a rule for advancing from one state to the next.

The hardest part of encoding this in Peano arithmetic is how to use an integer to encode a list of integers, and how to access / manipulate lists under this encoding.

(Then, once you have encoded the state of a computer as a single integer, you can then talk about the execution of a program on a computer as the "list of integers" corresponding to the state of the computer at a given time. With this, you can now start reasoning about how computers behave, again using just Peano arithmetic under-the-hood)


Formal logic is similar -- specifically look at the subfield of syntax, which talks about languages, grammar, and proofs. Here, you want to use a "list of integers" to store the characters in a formal expression -- i.e. to store strings.



There are other ways to manage this technical construction, but I think the isomorphism from your original post is the most straightforward. (it does require you to first prove some things about primes and divisibility, though)
 
  • #30
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So I typed it up again, generalizing [itex]\phi:\mathbb{Q}^{*}\to\mathbb{Z}^{\infty}[/itex].

I'm still a little confused how to turn this into a vector space. I was wondering, if we use the non-transcendental numbers as our field. But would we lose the fact of 'prime factorization', and the basis of even doing this?

I can't conceptually see how to work in [itex]\mathbb{Z}_{p}[/itex] for some prime [itex]p[/itex]. I don't know how we would chose [itex]p[/itex]. Maybe I'm just obsessing over minor points.

I've also defined [itex]\psi:\mathbb{Z}^{\infty}\to\mathbb{P}[X]^{\infty}[/itex], but I just don't know how to define a binary operation [itex]\otimes:\mathbb{Z}^{\infty}\times\mathbb{Z}^{ \infty}\to\mathbb{Z}^{\infty}[/itex] such that it is homomorphic by [itex]\psi[/itex] to multiplication of finite polynomials.
 
  • #31
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So I typed it up again, generalizing [itex]\phi:\mathbb{Q}^{*}\to\mathbb{Z}^{\infty}[/itex].

I'm still a little confused how to turn this into a vector space. I was wondering, if we use the non-transcendental numbers as our field. But would we lose the fact of 'prime factorization', and the basis of even doing this?

I can't conceptually see how to work in [itex]\mathbb{Z}_{p}[/itex] for some prime [itex]p[/itex]. I don't know how we would chose [itex]p[/itex]. Maybe I'm just obsessing over minor points.
Can't you just define it as a [itex]\mathbb{Q}[/itex]-vector space?

And maybe I'm starting to do much to advanced stuff here, but perhaps you could define your entire structure as a sheaf of rings??

I've also defined [itex]\psi:\mathbb{Z}^{\infty}\to\mathbb{P}[X]^{\infty}[/itex], but I just don't know how to define a binary operation [itex]\otimes:\mathbb{Z}^{\infty}\times\mathbb{Z}^{ \infty}\to\mathbb{Z}^{\infty}[/itex] such that it is homomorphic by [itex]\psi[/itex] to multiplication of finite polynomials.
You know that that is a bijection, right?? Well, then you can just define

[tex]a\otimes b=\psi^{-1}(\psi (a)\cdot \psi (b))[/tex]

just carry over the structure.

Do take a look at "group rings", because it really looks a lot like what you're trying to do here!
 
  • #32
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Can't you just define it as a [itex]\mathbb{Q}[/itex]-vector space?
That's what i mean, woops. But then we would be representing the non-transcendentals. Because if we choose [itex]\mathbb{Q}[/itex] to be our field, then since scalar multiplication in [itex]\mathbb{Z}^{\infty}[/itex] is isomorphic to exponentiation, things like [tex]\frac{1}{2}(1,0,\dots)\overset{\phi^{-1}}{=} 2^{\frac{1}{2}}=\sqrt{2}\notin\mathbb{Q}^{*}[/tex]So shouldn't we work with the set of non-zero algebraic numbers, as opposed to [itex]\mathbb{Q}^{*}[/itex] (this is what I meant as opposed to non-transcendental).

And maybe I'm starting to do much to advanced stuff here, but perhaps you could define your entire structure as a sheaf of rings??
Yeah, I haven't a clue what a sheaf of rings is >.<, i haven't even had a true introduction to group theory. I'm doing this a lot by reading as I go.



You know that that is a bijection, right?? Well, then you can just define

[tex]a\otimes b=\psi^{-1}(\psi (a)\cdot \psi (b))[/tex]

just carry over the structure.
Perfect! Can't believe I didn't think of that, thank you very much!

Do take a look at "group rings", because it really looks a lot like what you're trying to do here!
No I haven't yet, totally forgot you mentioned that. I'll definitely take a look now before I continue :)
 
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  • #33
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That's what i mean, woops. But then we would be representing the non-transcendentals. Because if we choose [itex]\mathbb{Q}[/itex] to be our field, then since scalar multiplication in [itex]\mathbb{Z}^{\infty}[/itex] is isomorphic to exponentiation, things like [tex]\frac{1}{2}(1,0,\dots)\overset{\phi^{-1}}{=} 2^{\frac{1}{2}}=\sqrt{2}\notin\mathbb{Q}^{*}[/tex]
Well, defining it like that isn't exactly what I mean :smile:

You know that [itex]\mathbb{Q}[/itex] is isomorphic to [itex]\mathbb{Z}^{\infty}[/itex] (if we adjoin a certain element 0 to the last set). So define scalar multiplication by

[tex]\alpha x=\varphi(\varphi^{-1}(\alpha)\varphi^{-1}(x))[/tex]. So just "lift" the scalar multiplication in [itex]\mathbb{Q}[/itex] to a new one in [itex]\mathbb{Z}^\infty[/itex].


Yeah, I haven't a clue what a sheaf of rings is >.<, i haven't even had a true introduction to group theory. I'm doing this a lot by reading as I go.
Well, I think that researching things because you need it in this problem is the only true way to learn something. So all power to you!!
But sheafs is too advanced :biggrin: It would work though.
 
  • #34
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Well, defining it like that isn't exactly what I mean :smile:

You know that [itex]\mathbb{Q}[/itex] is isomorphic to [itex]\mathbb{Z}^{\infty}[/itex] (if we adjoin a certain element 0 to the last set). So define scalar multiplication by

[tex]\alpha x=\varphi(\varphi^{-1}(\alpha)\varphi^{-1}(x))[/tex]. So just "lift" the scalar multiplication in [itex]\mathbb{Q}[/itex] to a new one in [itex]\mathbb{Z}^\infty[/itex].
I though I've already defined the scalar multiplication in [itex]\mathbb{Q}[/itex] in [itex]\mathbb{Z}^{\infty}[/itex] through defining [itex]\phi(a)=\phi(\prod^{\infty}_{k=1}p_{k}^{e_{k}})=(e_{1},e_{2},\dots)=(e_{k})^{\infty}_{k=1}[/itex]. And since [itex]\phi[/itex] is an isomorphism (I now show this properly in my paper), multiplication of numbers in [itex]\mathbb{Q}^{*}[/itex] is isomorphic to the 'addition' i defined ([itex]\oplus[/itex]) in [itex]\mathbb{Z}^{\infty}[/itex].

So there our vector addition is defined as multiplication in [itex]\mathbb{Q}^{*}[/itex] (or algebraic numbers, to generalize).

Then it makes sense that our scalar multiplication (from [itex]\mathbb{Q}[/itex]) would be isomorphic to exponentiation. Since it can be shown, for example, that

[tex]a^{2} \overset{\phi}{=} 2 (e_{k})^{\infty} _{k=1} = (e_{k}) ^{\infty} _{k=1} \oplus (e_{k}) ^{\infty}_{k=1}.[/tex]

I guess i'm actually generalizing even more instead of [itex]\phi:\mathbb{Q}^{*}\to\mathbb{Z}^{\infty}[/itex], but to [itex]\phi:\mathbb{A}^{*}\to\mathbb{Q}^{\infty}[/itex], where [itex]\mathbb{A}^{*}[/itex] is the set of all non-zero algebraic numbers.

Am I making sense or am I just crazy?
 
  • #35
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I though I've already defined the scalar multiplication in [itex]\mathbb{Q}[/itex] in [itex]\mathbb{Z}^{\infty}[/itex] through defining [itex]\phi(a)=\phi(\prod^{\infty}_{k=1}p_{k}^{e_{k}})=(e_{1},e_{2},\dots)=(e_{k})^{\infty}_{k=1}[/itex]. And since [itex]\phi[/itex] is an isomorphism (I now show this properly in my paper), multiplication of numbers in [itex]\mathbb{Q}^{*}[/itex] is isomorphic to the 'addition' i defined ([itex]\oplus[/itex]) in [itex]\mathbb{Z}^{\infty}[/itex].

So there our vector addition is defined as multiplication in [itex]\mathbb{Q}^{*}[/itex] (or algebraic numbers, to generalize).

Then it makes sense that our scalar multiplication (from [itex]\mathbb{Q}[/itex]) would be isomorphic to exponentiation. Since it can be shown, for example, that [tex]a^{2} \: \overset{\phi}{=} 2 (e_{k})^{\infty} _{k=1} = (e_{k}) ^{\infty) _{k=1} \oplus (e_{k}) ^{\infty )_{k=1}.[/tex]

I guess i'm actually generalizing even more instead of [itex]\phi:\mathbb{Q}^{*}\to\mathbb{Z}^{\infty}[/itex], but to [itex]\phi:\mathbb{A}^{*}\to\mathbb{Q}^{\infty}[/itex], where [itex]\mathbb{A}^{*}[/itex] is the set of all non-zero algebraic numbers.

Am I making sense or am I just crazy?
Hmm, I'm not sure how you would generalize all of this to algebraic numbers. But it does make sense what you're saying...
 
  • #36
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Hmm, I'm not sure how you would generalize all of this to algebraic numbers. But it does make sense what you're saying...
I guess I would I would have to prove a sort of analogous Fundamental Theorem of Arithmetic to the non-zero algebraics? If I can do that then I hope it all works, and I would just have to prove it to be a vector space, correct?

I never expected myself to get into doing this much for such a trivial idea I began with lol! Considering how little algebraic background I have outside of linear algebra.
 
  • #37
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I guess I would I would have to prove a sort of analogous Fundamental Theorem of Arithmetic to the non-zero algebraics? If I can do that then I hope it all works, and I would just have to prove it to be a vector space, correct?
I think such a thing would be very hard, if not impossible :frown:

I never expected myself to get into doing this much for such a trivial idea I began with lol! Considering how little algebraic background I have outside of linear algebra.
Hey, this is the perfect way to learn something more about algebra!! You're doing great!
 
  • #38
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As a last attempt, what if we just generalize to the set

[tex]\mathbb{A}^{*}=\{a^{b}:a,b\in\mathbb{Q}^{*}\}[/tex]

I would have to show this to be a field obviously.
 
  • #39
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As a last attempt, what if we just generalize to the set

[tex]\mathbb{A}^{*}=\{a^{b}:a,b\in\mathbb{Q}^{*}\}[/tex]

I would have to show this to be a field obviously.
That seems plausible. Don't know if it'll be a field, though...
 
  • #40
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That seems plausible. Don't know if it'll be a field, though...
I agree, I don't think its closed under addition.

I guess I never really tried [itex]\mathbb{Z}_{p}[/itex]. Or even the set [itex]\mathbb{Q}[\mathbb{Z}_{p}]=\{a^{b}|a,b\in\mathbb{Z}_{p}\}[/itex]. Which would be closed under addition (I'd hav to prove it) I guess since every [itex]a^{b}\equiv c<p[/itex], so we would have [itex]a^{b}+c^{d}\equiv e+f\equiv g[/itex] with [itex]e,f,g\leq p[/itex], or something like that.
 
  • #41
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So working i've got some weird results, that i'll type later when you fix a [itex]p>0[/itex] prime, and define [itex]\phi_{p}:\mathbb{Z}_{p}^{*}\to\mathbb{Z}_{p}^{n}[/itex] where [itex]1<p_{1}<p_{2}<\cdots <p_{n}<p[/itex] are all primes less than [itex]p[/itex].

I can show this to be a vector space, which is good. But you get weird equivalences when you work in [itex]\mathbb{Z}_{p}^{n}[/itex], that are just blowing my mind a little as I write them down on paper. Which get even weirder when you move to [itex]\mathbb{P}_{n} [\mathbb{Z}_{p}[/itex].

I'll type it all up when I get home.
 
  • #42
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Hmm, "weird equivalences", I'm wondering about what you will type up :smile:
 
  • #43
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Well its trivial to show that, from the same [itex]p[/itex] and [itex]n[/itex] defined before that [itex]\phi_{p}:\mathbb{Z}_{p}^{*}\to\mathbb{Z}^{n}_{p}[/itex] is an isomorphism. This is seen since if [itex]a\in\mathbb{Z}_{p}^{*}[/itex] then [itex]a[/itex] has a prime factorization such that [tex]a=\prod^{n}_{k=1}p_{k}^{e_{k}}=p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{n}^{e_{n}}.[/tex] So let [itex]\phi_{p}(a)=(e_{1},\dots,e_{n})[/itex]. Its really easy to show its a vector space over field [itex]\mathbb{Z}_{p}[/itex]

Whats interesting is when you partition the set such that for any [itex]u,v\in\mathbb{Z}_{p}^{n}[/itex], we have [tex]u\simeq v \;\; iff \;\; \phi_{p}^{-1}(u)\equiv\phi_{p}^{-1}(v).[/tex]

If you use the definition of the isomorphism i've used before into the polynomials, with the same partition, you'll get weird things. For example, take p=7. Then n=3. So we have [tex]0\simeq3\simeq x+x^{2}\simeq2+2x\simeq\cdots[/tex]

I've worked with quotient spaces in linear algebra, and I though how that collapsed a vector space was weird, but this weirds me out even more.

Its probably not weird for someone who has taken abstract"er" algebra than I have, but still weird lol

I'll be typing this all up sometime tonight or tomorrow when I have time in a paper and more rigourously, but i have other plans tongiht so I won't be able to type everything up i worked out today (like my proof that its a vector space).

I feel even that [itex]\mathbb{Z}_{p}^{n}[/itex] collapses [itex]\mathbb{Q}^{n}[/itex]. Since any n-tuple's entries can be seen as a quotient a/b, but then since [itex]\mathbb{Z}_{p}[/itex] is a field, so this quotient exists as one of the p members of Z_p.

Maybe thats wrong.
 
  • #44
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I don't quite see why [itex]\phi_p[/itex] is an isomorphism. That is, I see no reason why it should be surjective. Certainly something like (p-1,p-1,...,p-1) isn't reached?
 
  • #45
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I don't quite see why [itex]\phi_p[/itex] is an isomorphism. That is, I see no reason why it should be surjective. Certainly something like (p-1,p-1,...,p-1) isn't reached?
Sure it does, it corresponds to [itex]p_{1}^{p-1}p_{2}^{p-1}\cdots p_{n}^{p-1}\equiv a[/itex] with [itex]0\leq a\leq p-1[/itex].

Wait crap, that makes it non-injective. URGH.

I need to sit down with someone else and work all of this out. I need to get back to university asap. BLARGH.
 
  • #46
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Sure it does, it corresponds to [itex]p_{1}^{p-1}p_{2}^{p-1}\cdots p_{n}^{p-1}\equiv a[/itex] with [itex]0\leq a\leq p-1[/itex].

Wait crap, that makes it non-injective. URGH.

I need to sit down with someone else and work all of this out. I need to get back to university asap. BLARGH.
The problem is that you don't have a well-defined function, really.

[itex]\mathbb{Z}_p^*[/itex] has p-1 elements, and [itex]\mathbb{Z}_p^n[/itex] has [itex]p^n[/itex]elements. So there can never be a surjection. :frown:

Do you see a way to solve this??
 
  • #47
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Okay, I'm meeting with my prof tomorrow, and I want to really see if this works:

Define [itex]\mathbb{A}^{*} =\{a^{b}:a,b\in\mathbb{Q}^{*}\}[/itex] and let [itex]\phi :\mathbb{A}^{*}\to\mathbb{Q}^{\infty}[/itex], where [itex]\mathbb{Q}^{\infty}[/itex] is the set of all [itex]\infty[/itex]-tuples with a finite amount of non-zero entries. Note that for any [itex]a^{b}\in\mathbb{A}^{*}[/itex] we have a prime factorization such that [tex]a^{b}=\prod_{k=1}^{\infty}p_{k}^{be_{k}}.[/tex] So define [itex]\phi(a^{b})=(be_{1},be_{2},\dots)=(be_{k})^{\infty}_{k=1}[/itex]. Also define, from before, [itex]\oplus:\mathbb{Q}^{\infty}\times\mathbb{Q}^{\infty}\to \mathbb{Q}^{\infty}[/itex] such that [tex](e_{1},e_{2},\dots)\oplus (f_{1}, f_{2},\dots)=(e_{k})^{\infty}_{k=1}\oplus (f_{k})^{\infty}_{k=1}=(e_{k}+f_{k})^{\infty}_{k=1}=(e_{1}+f_{1},e_{2}+f_{2},\dots).[/tex] Further, [itex]\phi[/itex] can be shown to be an isomorphism.

We can also show [itex]\mathbb{Q}^ {\infty}[/itex] to be a rational vector space. Note that the scalar multiplication of any vector in [itex]\mathbb{Q}^{\infty}[/itex] is analogous to raising the original element of [itex]\mathbb{A}^{*}[/itex] by a rational exponent (the rational exponent being the scalar). That is: [tex]\alpha(be_{k})^{\infty}_{k=1}\overset{\phi^{-1}}{=}a^{b^{\alpha}}=a^{b\alpha}.[/tex] Also note that [itex]\alpha(be_{k})^{\infty}_{k=1}=\alpha b(e_{k})_{k=1}^{\infty}=(\alpha be_{k})^{\infty}_{k=1}[/itex].

I didn't want to type everything out, but it seems to have worked.
 
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  • #48
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So I've met with my professor, and by the end we just were laughing at how simple and familiar it ended up working out to be.

No matter how hard I tried, I could not create a vector space, which is fine. The suddenly it came to us.

So here:

Let [itex](\mathbb{R}^{>0},\cdot)[/itex] be the abelian group of positive reals and multiplication, and let [itex](\mathbb{R},+)[/itex] be the abelian group of the reals over addition. Suppose there exists a function [itex]\log:(\mathbb{R}^{>0},\cdot)\to(\mathbb{R},+)[/itex] such that [itex]\log(ab)=\log(a)+\log(b)[/itex] and [itex]\log(a^{b})=b\log(a)[/itex]. Its clear that [itex]\log[/itex] was just the extension of what I was trying to do before. I guess you could say its the 'analytic jump' from what I was trying to do algebraically. This has given me some new intuition into how [itex]\log[/itex] works. That its 'sort of' rooted in prime factorization, but more general than that.

I thought this was really interesting. And I'm glad this is all over of me attempting to find something that wasn't there, but had some sort of 'transcendental' version that explains everything.

He gave me some homework to do, the first question being one that I could probably do my self with my current skill sets, and another I'll have to wait after I take Measure Theory in the Winter Semester.

The first is that:

  • If [itex]f:(\mathbb{R}^{>0},\cdot)\to(\mathbb{R},+)[/itex] continuously with the properties that [itex]f(ab)=f(a)+f(b)[/itex] and [itex]f(a^{b})=b\cdot f(a)[/itex] and [itex]f(1)=0[/itex], then [itex]f(x)=\log(x)[/itex]

  • There exist many non-measurable [itex]f:(\mathbb{R}^{>0},\cdot)\to(\mathbb{R},+)[/itex] with the properties [itex]f(ab)=f(a)+f(b)[/itex], [itex]f(a^{b})=b\cdot f(a)[/itex], and [itex]f(1)=0[/itex].

I hope to be able to do these sometime in the future, but we'll see what happens.

Thank you to everyone that has helped me! Especially micromass for putting up with my lack of education in abstract algebra and metric space topology. I swear I'll get better after this year when I finally get a formal introduction to everything! haha
 
  • #49
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3,293
I'm very glad you have found an answer to your (very intriguing) question. I'm also happy that you learned a bit of new mathematics this way!!

I hope you enjoyed thinking about this. These are the things that math research is all about :smile:
 

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