Exponential fields as structures with three binary operations.

Question

The exponential rings and fields are usually studied as structures with two binary operations $(+,\cdot)$ and one unary operation $\exp(x)$ defined on a set $K$. Why not consider the exponential as a binary operation $\star : K\times K\rightarrow K\;,\quad x\star y=x^y$ and search for properties of a structure with three binary operations with suitable axioms? There is some subtle ''obstruction'' to study such kind of structures?

PS. I tag as a soft question because I search for a not too tecnical answer ( if possible).

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Added one month after. This question gained some reputation, but no answers nor comments. I read this result as : " the question seems interesting, but it's too hazy to talk about it". So I report here some my work, with the aim to better define the question.

The structure that i'm searching is $(E,+,\cdot,\star)$ where E is a not void set and:

Axiom 1) $(E,+,\cdot)$ is a ring with a multiplicative identity $1$. We assume that this ring has caracteristic $0$. (This in fact is a list of axioms).

In this ring we adopt the usual notations for $n\in \mathbb{N}$: $nx$ is the sum of $n$ identical elements and $x^n$ is the product of $n$ identical elements.

The binary operation $\star: E \times E \rightarrow E$ is such that, $ \forall x,y,z, \in E$ ve have:

Axiom 2) $\quad x\star (y+z)=(x\star y) \cdot (x\star z) $

From this we have : $x\star(ny)=(x \star y)^n$

Axiom 3) $\quad (x \cdot y)\star z= (x \star z) \cdot (y \star z) $

Axiom 4a) $ \quad 0 \star x=0 \qquad \forall x \in E$

Axiom 4b) $ \quad \forall x \ne 0 \Rightarrow \exists y \in E$ such that $x \star y \ne 0$

Axiom 4c) $ \quad \forall x\ne 1_E \Rightarrow \exists y \in E$ such that $x \star y \ne 1_E$

where $1_E$ is the neutral element of the product. These last two axioms are introduced to avoid triviality.

From these axioms we can proof some simple proposition, as:

Prop.1) $ \forall x,y,z, \in E$ we have:$(x\star y) \cdot (x\star z)=(x\star z) \cdot (x\star y)$.

Prop.2) $\forall x \in E$, if $(1_E-x\star 0)$ is a right zero divisor than it is also a left zero divisor and $x\star y$ is a zero divisor for all $y \in E$. If $(1_E-x\star 0)$ is not a zero divisor than $x \star 0=1 \; \forall x \in E$.

Prop.3) If $(1_E-x\star 0)$ is not a zero divisor the set $ R_x=\{y : \exists z \in E \rightarrow y=x \star z\}$ is a group under ring multiplication.

Prop 4) $\forall y \in E$, if $(1_E-1_E\star y)$ is a right zero divisor than it is also a left zero divisor and $x\star y$ is a zero divisor for all $x \in E$.

Prop. 5) If $(E,+,\cdot)$ is a division ring than $R=\bigcup_x R_x$ is a group under multiplication with $1_E$ as neutral element.

I stop here(for now). So my question is if these axioms are sufficient to define a ''nice'' structure, and if this approach can be interesting. Since my knowledge of universal algebra and category theory are very poor, I don't know if all this work has a sense, or if there are general results that can be used or that show that eventually this is an impasse.

Answer 1

In general, new structures are studied as the result of 1) realizing that multiple interesting objects possesses this structure, and 2) showing that this structure admits some nice results, and ideally helps tackle existing questions.

So if a certain structure isn't studied, either no one has thought to study it, or it doesn't have a nice set of examples or any obvious applicability, or (as is the case surprisingly often) the structure actually IS studied, and you didn't know about it because it isn't popular enough to be "mainstream".

Exponential fields may admit some interesting results, but first I'd like to know some motivating examples. When you're looking for examples it's good to know how to narrow your search, and for exponential fields it's easy to show that your axioms impose significant restrictions. For example since any exponential field $F$ is characteristic $0$, it must contain an isomorphic copy of $\mathbb{Q}$ -- and since you want the exponential operation $\star$ to be defined on the whole field and be compatible with exponentiation as iterated multiplication, since $1/n \in \mathbb{Q} \subset F$ for every $n \in \mathbb{N}$, that means the field has to be closed under $n^\textrm{th}$ roots, and so must be algebraically closed. That means the real numbers aren't even an example of an exponential field -- in particular, $-1 \star \frac{1}{2}$ isn't a real number!

Now don't let that discourage you -- it just means you probably have to look more deeply for examples. The complexes fit, and so do something called the complex hyperreals.

There is a very interesting field called the Hyperreals and written as $^*\mathbb{R}$, which is very similar to the real numbers (by which I mean, precisely, that it is an example of a real closed field, an object which has exactly the same "first order properties" as the reals). You can think of $^*\mathbb{R}$ as all real-valued sequences, but with some sequences considered to be "the same" in a way that makes the collection into a field. [E.g. the sequence $(0, 0, 0, \cdots)$ is considered to be the same as the sequence $(1, 0, 0, \cdots)$.]

The field $^*\mathbb{R}$ is not algebraically closed. Just like with the reals, you can form the algebraic closure by adjoining $\sqrt{-1}$. When you do, you get the complex hyperreals, written $^*\mathbb{C}$. This field is also an exponential field, since you can define

$$(r_1, r_2, \cdots)^{(s_1, s_2, \cdots)} = \left( r_1^{s_1}, r_2^{s_2}, \cdots \right) $$

and then prove that this does everything you need. (Warning: this is a bit delicate since you have to work with the equivalence relation that defines which sequences are "the same", and this equivalence relation is induced by an object called an ultrafilter, which can be a little cumbersome.)

So there are two examples -- the fields $\mathbb{C}$ and $^*\mathbb{C}$. But if your axioms only define two fields they probably aren't that interesting. What other examples can you find? (Also, if you relax the condition that the field be characteristic 0 you might find more examples, perhaps among the $p$-adics.) You may even stumble across objects that are almost examples but require a slight tweaking to your axioms. This would be good, since it would suggest you're on the right track!