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.

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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 ...
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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. ...
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Why do we stop at exponentiation stage in arithmetic of natural numbers?

In natural numbers the unary successor operator $S$ is the most natural function which maps each number to the next one. Furthermore we may consider the binary relation $+$ as an iteration of $S$. ...
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Superassociative operation

Background: Addition and multiplication are associative, but exponentiation is not. Question: Does an operation $\circ_1:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ exist such that ...
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Transfinite Knuth-arrow hierarchy vs. fast-growing hierarchy

Suppose Knuth arrow notation (and hence the hyperoperation sequence) is extended to transfinite ordinal indices as follows: Let μ be a large countable ordinal such that a fundamental sequence is ...
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Monotony of the function $a\uparrow ^b c$ in all the arguments

I want to prove the following monotony properties of hyper-operations : 1) $ a \uparrow^b c < (a+1) \uparrow^b c$ for all $a,b,c \ge 1$ 2) $ a \uparrow^b c < a \uparrow^b (c+1)$ for all $a\ge ...
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Is there any concept similar to unique factorization that applies to exponential operators?

We can talk about prime numbers over multiplication but is there any similar concept that applies to exponential operators or other hyperpowers like tetration? Can we use what we know about UFDs to ...
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Hyperoperations and $\mathsf{PA}$

I am very confused about the "role of the hyperoperations" in the peano arithmetic. For example addition's and multiplication's axioms are given. $A_1$ $\forall x(x+0=x)$ $A_2$ $\forall ...
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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 ...
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2answers
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definition of primes for higher hyperoperations

I was reading yesterday when I came across the history of counting. There was some evidence of an early understanding of prime numbers. I thought that I would try changing the definition of primality ...
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Example of Tetration in Natural Phenomena

Tetration is a natural extension of the concept of addition, multiplication, and exponentiation. It is quite obvious that there are things in the physics world which can be modeled by these 3 lowest ...
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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 ...
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Are points on the complex plane sufficient to solve every solvable equation composed of the hyperoperators, their inverses, and complex numbers?

Some background: I'm programming a maths environment. I'm computer science, so please excuse any probable ignorance and lack of precision in my question. It seems $i$ and complex numbers were ...
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Power tower inequality

I want to prove the following power tower inequality: $$ 3 \uparrow \uparrow 100 > 4 \uparrow \uparrow 99 $$ but I don't know how to do this. I think that induction will not work, because I think ...
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530 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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Order of Recursion?

Define an extended algebraic function $f(a)$ as a function on $a$ that utilizes any combination of recursive extensions and inverses of sequentiation. Example: $a + 1$ , sequentiation. $a + a$, ...
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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 ...
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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{^} ...
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Can exponentiation and power function be defined through Albert Bennett's operations?

In 1914 Albert Bennett suggested the following operation: $$a * b=a^0_2b=\exp(\ln a \ln b)$$ Now, given this function, addition and multiplication, and their properties, can one express ...
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2answers
286 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?
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2answers
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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"} \\ ...
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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 ...