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Questions tagged [hyperoperation]

Hyperoperation is a field of mathematics which studies indexed families of binary operations, Hyperoperations families, that generalize and extend the standard sequence of the basic arithmetic operations of addition, multiplication and exponentiation.

55
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12answers
7k views

Why are addition and multiplication commutative, but not exponentiation?

We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why. As background there are plenty of mathematical schemes that ...
5
votes
2answers
849 views

Example $x$, $y$ and $z$ values for $x\uparrow^\alpha y=z$ where $\alpha\in \Bbb R-\Bbb N$

$\uparrow^n$ and $G(n,\cdot,\cdot)$ are notations for hyperoperation. http://en.m.wikipedia.org/wiki/Hyperoperation $n$ is the hyperoperations rank. Can example $x$, $y$ and $z$ values be provided ...
4
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1answer
639 views

Where can I learn more about commutative hyperoperations?

I just learned about commutative hyperoperations, and they look interesting. However, the wikipedia page doesn't link to more information. Is there an article or book where I can learn more? I'm ...
23
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3answers
646 views

What combinatorial quantity the tetration of two natural numbers represents?

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. $$\...
21
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5answers
1k views

what operation repeated $n$ times results in the addition operator?

I had a difficult time in phrasing my question. But I was wondering if there is an operation that, when repeated n times, results in the addition operator. Same way as repeating addition n times ...
6
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3answers
472 views

Does anything precede incrementation in the operator “hierarchy?”

I here define the hierarchy of basic mathematical operators and their respective "inverse" operation (see hyperoperation). $$ \begin{array}{c|c|c|} & \text{Operator} & \text{"Inverse"} \\ \...
13
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2answers
1k views

Continuum between addition, multiplication and exponentiation?

I noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one. However, I was looking for the more general case where we find ...
8
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1answer
218 views

How to define $A\uparrow B$ with a universal property as well as $A\oplus B$, $A\times B$, $A^B$ in category theory?

In category theory there are definitions for $A\oplus B$, $A\times B$ and $A^B$ via universal properties. I wonder if it is possible to isolate a particular universal property to represent the ...
6
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2answers
477 views

Ordinal tetration: The issue of ${}^{\epsilon_0}\omega$

So in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next ...
8
votes
1answer
189 views

Are there nontrivial equations for hyperoperations above exponentiation?

A similar question was asked in comments elsewhere. A paper by Roberto Di Cosmo and Thomas Dufour ("The Equational Theory of 〈ℕ, 0, 1, + , ×, ↑〉 Is Decidable, but Not Finitely Axiomatisable") asserts ...
14
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2answers
1k views

How exactly does Knuth's Up-Arrow notation work?

I've done some research, and found this on Wikipedia. \begin{matrix}a\uparrow b=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix} \begin{matrix}a\...
16
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2answers
837 views

Could someone tell me how large this number is?

Context: If you guys didn't know, I'm running a nice little contest to see who can program the largest number. More specific rules if you are interested may be found in my chat room (click here to ...
7
votes
1answer
270 views

Algorithm for comparing the size of extremely large numbers

Is there a simple algorithm to decide which of the numbers $$a \uparrow ^b c \text{ and } d \uparrow ^e f$$ is the bigger one ? Using the hyperoperation, the numbers can be denoted with $$H_{b+2}(...
11
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1answer
703 views

Has this phenomenon been discovered and named?

If $$x-\frac{x}{2}=\frac{x}{2},$$ and $$\frac{x}{\sqrt{x}}=\sqrt{x},$$ and $$x-\uparrow(x-\uparrow^22)=x-\uparrow^22$$ when $(x\uparrow^n-[A])\uparrow^nA=x$, where $A$ is some constant, and one uses ...
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0answers
171 views

How to extend this extension of tetration? [closed]

if $0\le b<1$, then $a↑↑b = a^b$ if $b\ge1$, then $a↑↑b = a^{a↑↑(b-1)}$ if $b<0$, then $a↑↑b = \log_a(a↑↑(b+1))$ so for example, $2↑↑\pi = 2^{2^{2^{2^{0.1415926...}}}} = 21.5963561$ How can ...
6
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0answers
308 views

Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} (...
5
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2answers
343 views

Pentation Notation - How does it work? [duplicate]

When going through with learning Grahams number, I got stuck at $$3↑↑↑3$$ Working it through, we have $$3↑3=3^3$$ $$3↑↑3=3^{3^3}=3↑(3↑3)$$ As such, it would appear to me that $$3↑↑↑3=3^{3^{3^3}}=...
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2answers
309 views

Notation for function $ + \rightarrow \times $

Is there a standard notation to represent the function building the multiplication from the addition (I'm talking of the usual $+$ on $\Bbb N$)? I'm tempted by: $$ x \times y = + ^ y (x) $$ With ...
0
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2answers
80 views

There is a way to write TREE(3) via $F^a(n)$?

I read about Graham number and TREE(3). Graham number is: $f^{64}(4)$ where $f(n)=3\uparrow^n 3$ My question is: If there is a way to write ...
7
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2answers
275 views

Are hyperoperators primitive recursive?

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related ...
7
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3answers
696 views

Algorithm for tetration to work with floating point numbers

So far, I've figured out an algorithm for tetration that works. However, although the variable a can be floating or integer, unfortunately, the variable ...
4
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1answer
169 views

Comparing up-arrow's

Is it true that $$3\uparrow^{n+1} 3\ >\ n\uparrow^n n $$ holds for every $n\ge 1$ Since $3\uparrow^{n+1}3=3\uparrow ^n 3\uparrow ^n 3$ and $3\uparrow^n3$ is much bigger than $n$ for $n\ge 3$, ...
4
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2answers
578 views

What is the geometric, physical or other meaning of the tetration?

What is the geometric, physical or other meaning of the tetration or more high hyperoperations? Is it exists in general or it has only math concept?
4
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1answer
190 views

Tighter bounds on the fast growing hierarchy?

Not a dupe of this question, as I'm searching for tighter bounds. We define the fast growing hierarchy for finite values as follows: $$f_k(n)=\begin{cases}n+1,&k=0\\f_{k-1}^n(n),&k>0\end{...
3
votes
2answers
143 views

Is this a new operation? [duplicate]

I was thinking "What is before addition", and came up with this. This analogy describes it: Addition is to multiplication as [operation] is to addition. On wikipedia it says it is just "1+b". I came ...
1
vote
1answer
64 views

Hypersubstitution, m-ary terms, semigroups, equivalent definitions

A hypersubstitution $\sigma$ is (see, for example, Universal Algebra and Applications in Theoretical Computer Science, by Denecke and Wismath) mapping from term $f_i(x_1,...,x_{n_i})$ to the term $\...
1
vote
1answer
67 views

Hyperidentity, semigroups, bands.

Let a semigroup satisfy $F(x,x)\approx x$, where $F$ is a binary operation symbol.Let $B$ satisfy $x(yz)\approx (xy)z$ and $xx\approx x$. Does $B$ satisfy $F$ as a hyperidentity?We need only consider ...
0
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2answers
42 views

Are hyperoperations < 3 to a reciprocal of a positive integer equivalent to the 'root' inverse to that integer?

Can the logic of $\sqrt[n]x = x^{1/n}$ be applied to tetration and other natural numbered hyperoperations greater than exponentiation, or, do reciprocals of positive integers as the second argument of ...