Structures with addition, multiplication and exponentiation.

Question

The set $\mathbb{N}$ can be viewed as a mathematical structure with operations off addition, multiplication and exponentiation. Observe that:

1. It forms an Abelian monoid under both addition and multiplication.
2. Multiplication distributes over addition.
3. $1^x = x^{0} = 1$
4. $x^{a+b}=x^{a}x^{b}$
5. $x^{ab} = (x^a)^b$

Furthermore, the set $[0,\infty)$ can also be viewed in this way. Are there other interesting examples of this sort of thing? I am especially looking for examples lacking a natural total order.

Remark. A few more examples occur to me. However, they're both naturally ordered.

Firstly, for every strong limit cardinal $\kappa$, I think that the set $\{\nu < \kappa\}$ is an example of such a structure.

Secondly, if we drop the requirements that addition and multiplication be commutative, and require distributivity only on the left, as in $$x(a+b)=xa+xb,$$ then every non-trivial ordinal that is closed under exponentiation is an example of such a structure.

Answer 1

The integers modulo $n, \Bbb Z/n\Bbb Z$ are an example. One could argue whether they have a natural total order because $-1\equiv n-1 \pmod n$, but "natural" is a matter of taste.

Answer 2

There is such a thing called an Exponential Field, though the exponential here is more indicative of the unary function rather than a binary one, however. This is essentially precisely what you describe if we replace monoid with group. Since $\mathbb{N}$ and $[0,\infty)$ are semirings, I would guess the analogous term would be Exponential Semiring.

The complex numbers form such an exponential field, and they lack a total order.

Going with the Exponential semiring idea, I believe that you could take the semiring of $n\times n$ matrices with positive coefficients and use the exponential operator. This doesn't have the property that multiplication commutes or that $e^{A+B}=e^Ae^B$, however, since this requires the the matrices commute. This does not satisfy (5) either.

To correct this, we can take the semiring of all $n\times n$ diagonal matrices. In this case, we have commutative multiplication and addition, and $e^{A}$ is another diagonal matrix. This satisfies (5) when we examine the exponentiation as a power of a matrix rather than exponential, meaning that in your $x^{ab}=(x^a)^b$, we regard $b$ as a scalar.

To finally finish constructing an example, we can make the semiring of all $n\times n$ diagonal matrices with non-negative diagonal entries. We define an exponential operator that is binary by defining $\log(I+M)=\sum_{n\geq 1}{\frac{(-1)^{n-1}M^n}{n}}$ a la the Mercator series and defining $M^N=\exp(N\log M)$. This satisfies the properties $I^M=I=M^0$ and $M^{N+N'}=M^NM^{N'}$. To see that this satisfies (5), we notice that both the exponential and logarithm operation applied to a diagonal matrix are simply applications of the $\exp$ and $\log$ functions on the reals applied component-wise to the diagonal entries of the diagonal matrices. This is why we must take non-negative diagonal matrices (and positive diagonal matrices for the logarithm). When we do this, we find that it naturally fulfills (5). There is a slight-issue here though: this has a total ordering on it by giving it the dictionary ordering.

Perhaps taking complex diagonal matrices and taking principal values might give you what you're looking for, but there is a total order given on this as well by using the dictionary ordering as a total order on the complex numbers (with the natural ordering on $\mathbb{R}$) and then taking the dictionary ordering on the diagonal matrices.

Theoretically, since every set can be totally ordered, without making additional stipulations regarding what properties the order needs, not much can be said about exponential semirings with or without a total order. If we want that order to be compatible with the monoid operations, then you have a chance to make the object you want.

I'll update with examples as I think of them.

Answer 3

There's study of the system you describe under the name of Tarski's high school algebra problem. Tarski wrote down a system of 11 identities that correspond to your axioms, and asked if there were any identities that were true over the non-negative integers that couldn't be derived from those identities. The answer turns out to be yes -- there are true identities that cannot be proven. The link gives an explicit example.