# Sets as extremely trivial groups

A group is a structure defined upon an underlying set which is endowed with a single binary operator that has some rules attached to it. I was wondering whether one could describe a set itself as being a group defined upon itself such that the binary operator is "empty"/"null" and thus no real structure is established. To me, it makes some sense to build structures up by making the relations increasingly interesting: the most basic entity could be taken to be the set (endowed with the empty operator), and then a magma is pretty much the next most trivial, and then monoids, groups, rings, and fields emerge (in increasing order of "complexity" of structure in some sense; note that I skipped some stuff, ignored others, and eventually introduced multiple operators per structure).

On the other hand, I am having trouble conceptualizing sets to have even just the empty operator. Elements of sets do not interact with one another and even the empty operator would have to take an ordered pair of input and map it to... I am not sure what, maybe to "undefined" or something?

These two views seem to oppose each other. I prefer the former and view the latter as just some technical problems. But how can one be rigourous in this treatment?

I was wondering whether one could describe a set itself as being a group defined upon itself such that the binary operator is "empty"/"null" and thus no real structure is established.

We can do it; replace "group" with "structure".

See Jouko Vaananen, Models and Games (2011), page 54 :

Definition 5.1 An $\mathcal L$-structure (or $\mathcal L$-model) is a pair $\mathcal M = (M, Val_M)$, where $M$ is a non-empty set called the universe (or the domain) of $\mathcal M$, and $Val_M$ is a function defined on $\mathcal L$ with the following properties: [...].

If $\mathcal L = \emptyset$, an $\mathcal L$-structure $(M)$ is a structure with just the universe and no structure in it.

Since the set has a binary operation, it's signature is non-trivial, making it a magma. OTOH, a binary operation is, by definition a function

$$f: S\times S\to S$$

but it need not satisfy any axioms: that you can make empty. Magmas are really more of a categorifying term than an object of study, though--no structure means there's not much to talk about (even the wikipedia page is pretty empty), other than just a more specific kind of function (namely with a special kind of domain). Ultimately you know as much about a set with or without a binary operation if there's no axioms.

I would note that you cannot make your relation "empty," unless the set, $S$ is empty, because the definition of a function requires you have $f(x,y)$ defined for every $x,y\in S$, and although it can be anything it cannot be nothing.