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

= 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.

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)
 * 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: $$v_1\ne v_2\implies G_{v_1}\cap G_{v_2}=\emptyset$$.
 * 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 $$a_1a_2\in G_{v_1+v_2}$$.
 * 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 $$aa^{-1}=1$$ (here 1 is the 1 of $$G_0$$; that is, the real numbers).
 * 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. -- 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)