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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: $v_{1}\neq v_{2}\implies G_{v_{1}}\cap G_{v_{2}}=\emptyset$ .
The underlying set of the structure is the (disjoint) union of all the $G_{v}$ .
Require $G_{0}=\mathbf {R}$ , 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 $a\in G_{v}$ , $a\neq 0_{G_{v}}$ (here $0_{G_{v}}$ is the additive identity on $G_{v}$ ), then there exists $a^{-1}\in G_{-v}$ such that $aa^{-1}=1$ (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)[reply]

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)[reply]

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) [reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Notation

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

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

Resolved

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