First, let me state that I am not a formally-trained mathematician, therefore please forgive my approximations and help me correct my errors. Second, I am looking for prior work related to what I would like to call Hypertype Theory. Here is a short introduction to the topic.

The basic idea is this: a group is roughly defined with a triplet of two operators (one commutative and one that is not) and an identity term. A field is made of two such triplets plus a distributivity law. What if a third such triplet were to be added? And what if an unlimited number of such triplets could be added in a recursive manner? Would such an object exhibit any interesting properties?

Interestingly, there is one such object, and it can be constructed in a quite straightforward fashion by using Albert Bennett’s commutative hyperoperations. For readablity purposes, we introduce a new symbol for these binary commutative operators and for their non-commutative duals:

Commutative Operators

$a\overset{1}{\boxplus}b = e ^ {ln(a)} + e ^ {ln(b)} = a + b$

$a\overset{2}{\boxplus}b = e ^ {ln(a)} × e ^ {ln(b)} = a × b$

$a\overset{3}{\boxplus}b = e ^ {ln(a) × ln(b)} = a ^ {ln(b)} = b ^ {ln(a)}$

$a\overset{4}{\boxplus}b = e ^ {\displaystyle e ^ {ln(ln(a)) × ln(ln(b))}}$

$\cdots$

$\displaystyle a\overset{n+1}{\boxplus}b = e ^ {ln(a)\overset{n}{\boxplus}ln(b)}$

Non-Commutative Operators

$a\overset{1}{\boxminus}b = e ^ {ln(a)} - e ^ {ln(b)} = a - b$

$a\overset{2}{\boxminus}b = e ^ {ln(a)} ÷ e ^ {ln(b)} = a ÷ b$

$a\overset{3}{\boxminus}b = e ^ {ln(a) ÷ ln(b)} = a ^ {1 ÷ ln(b)}$

$a\overset{4}{\boxminus}b = e ^ {\displaystyle e ^ {ln(ln(a)) ÷ ln(1 ÷ ln(b))}}$

$\cdots$

$a\overset{n+1}{\boxminus}b = e ^ {ln(a)\overset{n}{\boxminus}ln(b)}$

Identity Terms

$i_1 = 0$

$i_2 = 1$

$i_3 = e$

$i_4 = e ^ e$

$\cdots$

$i_{n+1} = e ^ {i_n} = e \upuparrows (n - 1) \quad (n > 1)$

Core Identities

From these definitions, we can recursively prove the following core identities:

$a\overset{n}{\boxminus}i_n = i_n$

$a = b \Longleftrightarrow a\overset{n}{\boxminus}b = i_n$

$a\overset{n}{\boxminus}(b\overset{n}{\boxminus}c) = c\overset{n}{\boxminus}(b\overset{n}{\boxminus}a)$

$(a\overset{n}{\boxplus}b)\overset{n}{\boxminus}c = a\overset{n}{\boxminus}(c\overset{n}{\boxminus}b)$

In general, these core identities can be used as axiomatic definitions, but it just so happens that we can prove them when the operators are defined using commutative hyperoperators. This ensures that we are not dealing with purely-theoretical objects.

Group Properties

From the core identities stated above, we can prove the following properties:

$\overset{n}{\boxminus}$ Anticommutativity

$a\overset{n}{\boxminus}b = i_n\overset{n}{\boxminus}(b\overset{n}{\boxminus}a)$

$\overset{n}{\boxplus}$ Left and Right Identity

$i_n\overset{n}{\boxplus}a = a\overset{n}{\boxplus}i_n = a$

$\overset{n}{\boxplus}$ Left and Right Identity

$i_n\overset{n}{\boxplus}a = a\overset{n}{\boxplus}i_n = a$

$\overset{n}{\boxplus}$ Commutativity

$a\overset{n}{\boxplus}b = b\overset{n}{\boxplus}a$

Additional Properties

Along the way, the following properties can be established as well:

Double Reverse

$i_n\overset{n}{\boxminus}(i_n\overset{n}{\boxminus}a) = a$

Associative Commutativity

$(a\overset{n}{\boxminus}b)\overset{n}{\boxminus}c = (a\overset{n}{\boxminus}c)\overset{n}{\boxminus}b$

Affine Equivalence

$a\overset{n}{\boxminus}b = c \Longleftrightarrow a\overset{n}{\boxminus}c = b$

Dual Substitution

$a\overset{n}{\boxplus}b = a\overset{n}{\boxminus}(i_n\overset{n}{\boxminus}b)$

Dual Identities

$(a\overset{n}{\boxminus}b)\overset{n}{\boxplus}b = a$

$(a\overset{n}{\boxplus}b)\overset{n}{\boxminus}b = a$

Hypertypes

We define a sequence of hypertypes by granting the hypertype $\mathbb{H}_n$ the triplet $(\overset{n}{\boxplus}, \overset{n}{\boxminus}, i_n)$ and all the triplets granted to its predecessors. This makes $\mathbb{H}_1$ isomorphic to $\mathbb{Z}$ and $\mathbb{H}_2$ isomorphic to $\mathbb{Q}$, while $\mathbb{H}_3$ is isomorphic to a strict subset of $\mathbb{R}$, which we call exponential numbers $\mathbb{E}$.

We have yet to study distributivity properties on hypertypes for $n > 2$.

We call our structure Hypertypes instead of Hypergroups for three main reasons: first, Hypergroups refer to an already-existing collection of objects that have nothing to do with Hypertypes; second, we are interested in developing the Hypertype Theory in the context of Type Theory, without relying on any of the axioms of Set Theory; third, we are interested in treating commutative operators separately from their non-commutative duals.

The third point is of critical importance: in order to properly handle real-world measures like temperatures that are non-additive (physicists and statisticians call them intensive), we must be able to distinguish them from measures that are additive (extensive). Therefore, our hierarchy of types should add one operator at a time, always starting with the non-commutative operator. Therefore, for every hypertype $\mathbb{H}_n$, we will have an intensive sub-hypertype $\mathbb{H}_{\bar{n}}$ defined with the non-commutative operator but without its commutative dual.

This approach is nicely supported by the fact that the Core Properties defined above are written with affine-style identites focused on the non-commutative operator. Therefore, we can properly define the commutative operator from its non-commutative dual. This approach also reduces the number of axiomatic definitions that need to be stated when defining the two operators.

Notation

The notation introduced above is especially useful for $n = 3$. Therefore, we suggest that the index $n$ can be ommitted whenever it is equal to $3$. Doing so, we would benefit from a single-symbol notation for the commutative power law and its inverse.

$a \boxplus b = a\overset{3}{\boxplus}b = e ^ {ln(a) × ln(b)} = a ^ {ln(b)} = b ^ {ln(a)}$

$a \boxminus b = a\overset{3}{\boxminus}b = e ^ {\frac{ln(a)}{ln(b)}} = a ^ {\frac{1}{ln(b)}}$

Alternatively, if we were to limit ourselves to symbols found on most keyboards, we would suggest:

$a \# b = a\overset{3}{\boxplus}b = e ^ {ln(a) × ln(b)} = a ^ {ln(b)} = b ^ {ln(a)}$

$a \backslash b = a\overset{3}{\boxminus}b = e ^ {\frac{ln(a)}{ln(b)}} = a ^ {\frac{1}{ln(b)}}$

Random Thoughts

What follows is not yet related to the hypertypes introduced above, but will most certainly support some future work on the topic. One of the main reasons for introducing the notion of hypertype is the realization that not all irrational numbers should be treated the same. For example, $e$ is very naturally introduced by $\mathbb{H}_3$, but it is quite unlikely that $\pi$ will ever find its place within any $\mathbb{H}_n$.

The reason for this is that $\pi$ only makes sense when a second dimension is added. Within $\mathbb{H}_3 \times \mathbb{H}_3$, Euler’s identity $e ^ {i\pi} + 1 = 0$ (probably the most beautiful identity of all mathematics) brings it to the fore very naturally. In other words, when we consider $\pi$ within the field of real numbers $\mathbb{R}$, we’re only looking at a one-dimensional projection of what is a fundamentally two-dimensional object, unlike many other irrational numbers such as $e$ and all other exponential numbers, which are fundamentally one-dimensional. In other words, $e$ is much simpler than $\pi$.

Another way to think about this is to realize that the gap between $\mathbb{Q}$ and $\mathbb{R}$ is much bigger than we like to think when we build the latter by completion of the former. A significant chunk of that gap can be filled by successive hypertypes, but another chunk should really be studied within the context of projecting multi-dimensional objects onto the continuous line, which is a poor approximation.

Therefore, the purpose of this work is to provide a simple framework that can be used to distinguish one-dimensional irrational numbers from other, more complex irrational numbers.

Request

If any of this is of any value, please let me know, because I really have no clue.

Credits

Many thanks to the following people for their contributions:

• @Henry for having found the triplet for $\mathbb{H}_3$.
• Prof N J Wildberger for being such an inspiration to young contrarian mathematicians.