How could we define negative hyper operators?
#1
So, hyper operators are defined recursively within all natural numbers, which makes them easy to understand although hard to extend (tetration and more). But how could we define negative hyper operators? Like the inverse function of their opposites? Like subtraction for the -1th hyperoperator? What about -3? Do we take the log, root, both, or something else?
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#2
tl.dr: the question is too vague. It depends on the definition of hyperoperations you are currently working on.

Full answer: I'll use Robbins' working definition of hyperoperations. Let \(I\) be a set of indexes, or ranks. Let \(A\) be a number system, eg. naturals, integers, rationals, reals or complex rings. Given a map \(H:I\to {\rm Op}_2(A)\) that associates to each index a binary operation \(A\times A\to A\) over our chosen number system \(A\). We call that \(H\) an hyperoperations sequence/family iff there exists three distinguished indexes/ranks \(i_1,i_2,i_3\in I\) s.t

$$H_{i_1}=+_A\quad\quad H_{i_2}=\times_A\quad\quad H_{i_3}=\exp_A$$

There are tons of models of this definition: Goodstein hyperoperations, Knuth notation, lower hos, balanced hos, Bennet's commutative operations (this is a model in a weak sense) and so on.

I know, this seems a bit vague but, trust me, it can be made formal. Also not all the models are built recursively in the classical sense: Bennet's one is defined half-recusively, half analytically (given a superfunction of exp).

Your question can be rephrased as follows. Let \(\mathbb N= I\) and given an \(H:\mathbb N\to {\rm Op}_2(A)\) that is hyperoperative in the previous sense, i.e. it interpolates addition, multiplication and exponentiation, how can we define an hyperoperative map \(H':\mathbb Z\to {\rm Op}_2(A')\) that extends \(H\)?

Some cases are known.
-Bennet operators: in this case it is perfectly possible to define operations for all the negative indexes by iterated application of log/exp conjugation. This was investigated by Rubtsov and many times here in this forum. Te obtained continuum of operations is made of isomorphic copies of the field of real numbers but "log-scaled". It was claimed by Rubtsov that this system of fields can be merged into a single universal fields, called delta field, featuring all the formal inverses of every exotic subtraction. I was able to formally define a finite portion of it in some unpublished notes but the complete construction is still eluding me: I claim it can be easily reached by categorical means through some direct categorical limit construction.

-Goodstein operations: this case is settled as trivial. Asking for the defining equation to hold universally implies zeration triviality, and triviality of all the negative ranks. This is known since the age of the Zeration thread by Trappmann and other user and this is the main reason the discussion on choosing different criteria of extension arose (see Mother Laws). For more see all the references at [2015, p2, remark 1]. Also, the triviality of all the negative rank is about subfunction operator having a fixed point at the successor function: this was already noted by Nixon (JmsNxn) and Raes (Tommy) back in [2011] and recognized as possible obstruction to having genuine interesting rational ranks, see [2015b].

-Non-trivial-zeration extensions: in this case you better study  Rubtsov and Romerio's proposals, Cesco Reale's and some of the references I gave in the previous point. If you don't ask for the law to hold universally, you restrict to \({\rm Op}_2(\mathbb N)\), then it is perfectly possible to have non-trivial negative ranks because the recursion only computes the negative ranks operation in a partial subsets leaving the rest of the values to be up to choice [2015 Tencer]. In an unpublished paper in italian I showed how most of the possible solutions of that kind can be defined in term of max-minus operations yelding an interesting link to tropical mathematics. For more see [2014]

I don't know much more about the extension of other families since I had not the time to study them. I believe that all the recursion-based families will suffer similar obstruction caused by very general phenomena related to what I call theory of Goodstein ranks. In short, this has to do with the admissible chain of consecutive solutions to conjugacy-like equations inside non-commutative monoids.


More considerations can be made on the meaning of going backwards in this case. This kind of approach, called negative thinking, is often fruitful but can be carried in many different ways.
For example, we could instead focus on distributivity as a trait d'union between three initial ranks. This will likely make us miss tetration as a fourth step, but will give us an interesting view. The pattern is the following
\((a*_0 b)+k=(a+k)*_0(b+k)\)
\((a+b)k=ak+nk\)
\((ab)^k=a^kb^k\)
\((a^b)*_4 k=(a*_4 k)^{(b *_4 k)}\)
In general we could ask for a sequence of algebraic structures \(A_i:=(A,*_i)\) over the same support set \(A\) equipped with set maps \(\phi_i:A\to {End}(A_i)\) such that $$\phi_i=\rho^{*_{i+1}}$$ i.e. \(\phi_i\) coincides the right translation map \(\rho^{*_{i+1}}:A\to A^A\) of the \(i+1\)-th operation. $$\forall k\in A.\, \phi_i(k)=\rho^{*_{i+1}}_k: a \mapsto a*_{i+1} k$$

At that point, with negative thinking, we could just ask for all the possible algebraic structures \(Z=(A,\bullet)\) over \(A\) with the property that the map \(\rho^{*_{1}}:A\to A^A\) has its output inside the subset \({\rm End}(Z)\subseteq A^A\).
In this example we have that the min and the max operations are possible solution to this backward problem.

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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#3
God! I always love me a good Mphlee post. You've really progressed as a mathematician, and your referencing is highly fascinating.

I'd also like to add, as an aside, more than anything, that any negative rank in an analytic sense results in singular behaviour. Any attempt to write:

$$
a[s]b\\
$$

For \(a[0]b = a+b,\,a[1]b = ab,\,a[2]b = a^b\), while still satisfying goodstein's equation--results in:

$$
a[-1] b = a+1\\
$$

But \(s \approx -1\) displays strong singular behaviour. So when defining "analytic semi-operators", the negative integers act as singularities, in and of themself, where they look like successorship \(a[-2]b = a+1\), but are actually very weird singularities. Where as something like \(a [-1.5] b\) is plausible, and can be analytic, you won't see this at the negative integers themself.

And before we get into analytic semi-operators, I believe I have finalized the proof that Bennet's operations can be massaged to produce Goodstein's equation. Again though, I cannot run code (I've given up on programming it). But the math looks 100%, as the kids say.
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