Testwiki:Reference desk/Archives/Mathematics/2023 October 26

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October 26

A particular type of algebraic structure

I'm curious whether anyone has seen this sort of algebraic structure, or knows a name for it. (I'm also curious how many people will figure out what I have in mind.) Some of the details are a little to be fleshed out.

• Start with a real vector space $V$. (Actually a Z-module might be sufficient, but I don't see any use for the extra generality and some things are a little easier if I have a vector space.)
• Now, for each $v\in V$, fix a nontrivial additive abelian group $G_v$, and these must be pairwise disjoint: $v_1\ne v_2⟹G_{v_1}\cap G_{v_2}=\mathrm{\varnothing }$.
• The underlying set of the structure is the (disjoint) union of all the $G_v$.
• Require $G_0=𝐑$, the real numbers. (Here 0 is the zero vector of $V$.)
• Addition is defined only between elements of the same $G_v$.
• Multiplication, however, is defined on the whole structure, and if $a_1\in G_{v_1}$ and $a_2\in G_{v_2}$, then $a_1{a}_2\in G_{v_1+v_2}$.
• Multiplication is commutative and associative.
• Multiplication is distributive when all terms are defined; that is, for any $v_1,v_2\in V$, $a\in G_{v_1}$, $b,c\in G_{v_2}$, we have $a(b+c)=ab+ac$.
• Nonzero elements have multiplicative inverses: If $a\in G_v$, $a\ne 0_{G_v}$ (here $0_{G_v}$ is the additive identity on $G_v$), then there exists $a^{-1}\in G_{-v}$ such that $a{a}^{-1}=1$ (here 1 is the 1 of $G_0$; that is, the real numbers).

I think that's it! Anyone know what such a gadget is called, maybe after tweaking one or two of my slightly arbitrary choices? Anyone see what I'm getting at? --Trovatore (talk) 03:23, 26 October 2023 (UTC)

Can we not extend addition to the whole thing by considering it as addition on the direct sum of all the G's? If so, then we just have a really big field.--Jasper Deng (talk) 06:24, 26 October 2023 (UTC)

That would be a bigger structure, I think. I really just want the disjoint union of all the $G_v$, not some bigger thing generated by them. --Trovatore (talk) 07:09, 26 October 2023 (UTC) Actually, I also don't see why it should be a field. How are you going to get multiplicative inverses of sums of elements from different $G_v$'s? --Trovatore (talk) 07:34, 26 October 2023 (UTC)

By the way, on reflection, I think I do want to say for now that $V$ is a free Z-module rather than a real vector space. I have in mind a related version where it should be a vector space, but it introduces another complication I hadn't noticed. --Trovatore (talk) 07:15, 26 October 2023 (UTC)

Does the construction require more of $V$ than it being an abelian group? Also, why does the second step require the assigned groups $G_v$ to be nontrivial?  --Lambiam 08:15, 26 October 2023 (UTC)

You can make the same definition without these requirements, but it would allow models that don't look like what I have in mind. --Trovatore (talk) 16:01, 26 October 2023 (UTC)

OK, I'll spoil the riddle. My thought is that these structures, or something similar to them, constitute a natural setting for dimensional analysis. Each element $v\in V$ is the dimensions of some type of quantity; an element of $G_v$ is a quantity having those dimensions.

A "coherent system of units" is a basis $B$ for $V$ together with, for every $b\in B$, a distinguished nonzero element $u_b\in G_b$. Any element in the disjoint union can be expressed uniquely up to mumble mumble by a real number times a product of integer powers of finitely many of the $u_b$.

I think the strength of this approach is in what it forgets. There is no preferred system of units. There's not even a preferred set of fundamental dimensions. If you think the basic dimensions are length, time, and mass, and I think they're length, time, and force, that's just fine. We can work in the exact same structure with the exact same quantities. We're just using a different basis for $V$.

If I were naming this myself, perhaps I'd call it a "dimensional system". But my guess is that someone as already isolated this concept (or something very similar).

With that hint, does anyone know? --Trovatore (talk) 19:37, 26 October 2023 (UTC)

Spoiling it further, Terry Tao has a nice blog on the subject and this paper may also be of interest. --{{u|Mark viking}} {Talk} 22:12, 26 October 2023 (UTC)

TAOOOOOO!!. Thanks. The second paper touches on something I was thinking about when I changed the specification of $V$ from a vector space to a free Z-module. To make use of a vector space you want to be able to take fractional powers of quantities, but only the positive ones keepin' it real so you have to make the $G_v$'s into ordered groups or something, require that the order play nice with the multiplication, etc. One difference I see in these expositions is that (based on my very quick scan; I could be wrong) it looks like they start with fundamental dimensions (length, time, etc), and then maybe they wind up with something where you can change the basis, but I start with no preferred basis. --Trovatore (talk) 01:01, 27 October 2023 (UTC)

Notation

Is $\mathbb{Z}/2\mathbb{Z}$ the integers modulo 2? Bubba73 ^{You talkin' to me?} 06:05, 26 October 2023 (UTC)

Yes.--Jasper Deng (talk) 06:25, 26 October 2023 (UTC)

Thanks. Bubba73 ^{You talkin' to me?} 06:36, 26 October 2023 (UTC)