Hyperoperations Wiki - User contributions [en]

Fractional iterate

2026-01-02T15:48:41Z

MphLee:

$\phi$ is called an $\frac{m}{n}$-iterate of $f$ iff
$$\phi^n(x) = f^m(x)$$

Equivariant map

2025-11-12T09:36:51Z

MphLee:

Let $M$ be a monoid and $X,Y$ sets. A function $\phi:X\to Y$ from the set $X$ to the set $Y$ is defined to be '''$M$-equivariant''' if all the following conditions hold

<ol style="list-style-type:lower-roman">
<li> exists a left $M$-action $f:M\times X \to X$ on the set $X$;
<li> exists a left $M$-action $g:M\times Y \to Y$ on the set $Y$;
<li> (equivariance) for all $t\in M$, for all $x\in X$, the equation $\phi(f(t,x))=g(t,\phi(x))$ holds.
</ol>

We denote by $M{\rm Set}((X,f),(Y,g) )$ the set of $M$-equivariant maps from the M-Sets $(X,f)$ and $(Y,g)$. It is a subset of the set $Y^X$ of all the set-theoretic functions from $X$ to $Y$.

'''Notation''' From now we use the following abuse of notation: we use $X$ to denote the left action $(X,f)$, i.e. the '''set equipped with the action''', the infix notation for the action $tx:=f(t,x)$ and denote by $M{\rm Set}(X,Y )$ the set $M{\rm Set}((X,f),(Y,g) )$.

==Particular cases==
Some special cases are [[Abel function]], [[Schröder coordinates]], [[Böttcher coordinate]], some [[primitive recursive functions]] and [[linear transformations]] of [[vector spaces]].
The concept is very flexible since every function $\phi:x\to Y$ can always bee seen as $M$-equivariant with respect to the [[constant action]], i.e. the action of ''doing-nothing'', on the sets $X$ and $Y$.

==General case==
Equivariant maps are a special kind of [[natural transofrmations]] $\phi:X\to Y$ between two [[functors]] $X,Y: {\bf B}M \to {\rm Set}$ going from the monoid $M$, seen as a single-object category, to the category of sets or other categories.

==Properties==
Let $\phi$ be $M$-equivariant, $x_0,x_1 \in X$ and $\lambda\in {\rm ht}_X(x_0,x_1):=\{\lambda\in M\,|\, \lambda x_0=x_1\}$. If $\phi(x_0)=\phi(x_1)=y$ then $\lambda$ is a [[period]] of $y\in Y$.

'''proof'''. This follows from the [[Grothendieck construction]] $\int:M{\rm Set}\to {\rm Cat}/_{{\bf B}M }$, being a functor: each $\phi\in M{\rm Set}(X,Y )$ is lifted to a functor $\int\phi\in {\rm Cat}/_{{\bf B}M}(\int X,\int Y )$. Since the height-sets ${\rm ht}_X(x_0,x_1)$ in $X$ are in bijection with the homsets in $\int X$, i.e. ${\rm ht}_X(x_0,x_1)\cong \int X(x_0,x_1)$ the functor $\int\phi:\int X\to \int Y$ induces a family of set-maps ${\rm ht}_X(x_0,x_1)\to {\rm ht}_Y(\phi(x_0),\phi(x_1))$. By assumption $\phi(x_0)=\phi(x_1)=y$ we induce an inclusion map $${\rm ht}_X(x_0,x_1)\subseteq {\rm per}_Y(y) $$

Equivariant map

2025-11-12T08:34:11Z

MphLee:

Let $M$ be a monoid a,d $X,Y$ two sets. A function $\phi:X\to Y$ from the set $X$ to the set $Y$ is defined to be '''$M$-equivariant''' if all the following conditions hold

<ol style="list-style-type:lower-roman">
<li> exists an $M$-action $f:M\times X \to X$ on the set $X$;
<li> exists an $M$-action $g:M\times Y \to Y$ on the set $Y$;
<li> (equivariance) for all $t\in M$, for all $x\in X$, the equation $\phi(f(t,x))=g(t,\phi(x))$ holds.
</ol>

We denote by $M{\rm Set}((X,f),(Y,g) )$ the set of $M$-equivariant maps from the M-Sets $(X,f)$ and $(Y,g)$. It is a subset of the set $Y^X$ of all the set-theoretic functions from $X$ to $Y$.

==Particular cases==
Some special cases are [[Abel function]], [[Schröder coordinates]], [[Böttcher coordinate]], some [[primitive recursive functions]] and [[linear transformations]] of [[vector spaces]].
The concept is very flexible since every function $\phi:x\to Y$ can always bee seen as $M$-equivariant with respect to the [[constant action]], i.e. the action of ''doing-nothing'', on the sets $X$ and $Y$.

==General case==
Equivariant maps are a special kind of [[natural transofrmations]] $\phi:X\to Y$ between two [[functors]] $X,Y: {\bf B}M \to {\rm Set}$ going from the monoid $M$, seen as a single-object category, to the category of sets or other categories.

Equivariant map

2025-11-12T08:31:31Z

MphLee: Created page with "Let $M$ be a monoid a,d $X,Y$ two sets. A function $\phi:X\to Y$ from the set $X$ tot he set $Y$ is defined to be '''$M$-equivariant''' if all the following condit..."

Let $M$ be a monoid a,d $X,Y$ two sets. A function $\phi:X\to Y$ from the set $X$ tot he set $Y$ is defined to be '''$M$-equivariant''' if all the following conditions holds

<ol style="list-style-type:lower-roman">
<li> exists an $M$-action $f:M\times X \to X$ on the set $X$;
<li> exists an $M$-action $g:M\times Y \to Y$ on the set $Y$;
<li> (equivariance) for all $t\in M$, for all $x\in X$ the equation $\phi(f(t,x))=g(t,\phi(x))$ holds.
</ol>

We denote by $M{\rm Set}((X,f),(Y,g) )$ the set of $M$-equivariant maps from the M-Sets $(X,f)$ and $(Y,g)$. It is a subset of the set $Y^X$ of all the set-theoretic function from $X$ to $Y$.

==Particular cases==
Some special cases are [[Abel function]], [[Schröder coordinates]], [[Böttcher coordinate]], some [[primitive recursive functions]] and [[linear transformations]] of [[vecvtor spaces]].
The concept is very flexible since every function $\phi:x\to Y$ can bee seen as $M$-equivariant with respect to the [[constant action]], i.e. the action of ''doing-nothing'', on the sets $X$ and $Y$.

==General case==
Equivariant maps are a special kind of [[natural transofrmations]] $\phi:X\to Y$ between two [[functors]] $X,Y: {\bf B}M \to {\rm Set}$ going from the monoid $M$, seen as a single-object category, to the category of sets or other categories.

Principal Abel function

2025-11-12T07:46:49Z

MphLee:

The principal Abel function (up to an additive constant) of $f$ at fixpoint $a$ is the [[Abel function]] of $f$ such that the [[fractional iterate]]s $\phi_c$ defined by
$$\phi_c(x) = \alpha^{-1}(c+\alpha(x)), c\in\Q$$
are [[regular iterate|regular]] at the fixpoint $a$.

Regular iteration

2025-11-12T07:46:03Z

MphLee:

A [[fractional iterate]] $\phi$ of an analytic function $f$ at fixpoint $a$ is called regular, iff $\phi$ is analytic at $a$ or has an [[asymptotic powerseries development]] at $a$.

Template:Navigation box test

2025-05-22T16:00:34Z

MphLee:

{| class="wikitable"
|-
! style="background-color:#6665cd;" |
! style="background-color:#9698ed;" | kind 1
! colspan="2" style="background-color:#9698ed;" | kind 2
|- style="background-color:#cbcefb;"
| style="background-color:#9698ed;" | object 1
| TEST
| item 1
| item 2
|-
| style="background-color:#9698ed;" | object 1
| [[Tetration]] - [[Pentation]]
| [[iteration]]
|
|-
| style="background-color:#9698ed;" | object 2
| $x^{x+1}$ - $H_5(z+1)={}^{H_5(z)}b$
|
|
|}

Template:Navigation box test

2025-05-22T15:58:04Z

MphLee:

Test

2025-05-22T15:52:32Z

MphLee:

This page is for any testing that you want to do.

$x\in\R$ huhu and $z\in\C$

=Section=
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

==Subsection==
Lorem ipsum

===1. subsubsection===
Lorem ipsum

===2. subsubsection===
Lorem ipsum

====Test====

For creating tables use [https://www.tablesgenerator.com/mediawiki_tables# Tables Generator]

{{Navigation box test}}

Template:Navigation box test

2025-05-22T15:28:43Z

MphLee:

{{Navbox
| name = Navigation box test{{subst:void|Don't change anything on this line. It will change itself when you save.}}
| title =
| listclass = hlist
| state = {{{state|}}}

| above =
| image =

| group1 =
| list1 =

| group2 =
| list2 =

| group3 =
| list3 =



| below =
}}<noinclude>
{{navbox documentation}}

</noinclude>

Template:Navigation box test

2025-05-22T15:18:25Z

MphLee: Created page with "{{Navbox |name = |state = |title = |image = |group1 = |list1 = }}"

{{Navbox
|name =
|state =
|title =
|image =
|group1 =
|list1 =
}}

Test

2025-05-22T14:40:58Z

MphLee:

Template:Navbox test

2025-05-22T14:23:39Z

MphLee: Replaced content with "{{Navbox}}"

Template:Navbox test

2025-05-22T14:18:37Z

MphLee:

{{Navbox
| title = Test
| name = fn org
| above = Above
| below = Below
| group1= first group
| list1 = [[List1]]
| list2 = [[List2]]
| list3 = [[List3]]
}}

Category:Pages with template loops

2025-05-22T13:44:44Z

MphLee: Created page with "test navbox"

test navbox

Template:Navbox test

2025-05-22T13:41:59Z

MphLee:

{{Navbox
| name=test
| title=[[Hyperoperation|Hyperoperations]]
| listclass=hlist
| group1 = Primary
| list1 =
*[[Successor function|Successor (0)]]
*[[Addition|Addition (1)]]
*[[Multiplication|Multiplication (2)]]
*[[Exponentiation|Exponentiation (3)]]
*[[Tetration|Tetration (4)]]
*[[Pentation|Pentation (5)]]
*[[Hexation|Hexation (6)]]
| group2 = [[Inverse function|Inverse]] for left argument
| list2 =
*[[Primitive_recursive_function#Predecessor|Predecessor (0)]]
*[[Subtraction|Subtraction (1)]]
*[[Division (mathematics)|Division (2)]]
*[[nth root|Root extraction (3)]]
*[[Super-root|Super-root (4)]]
| group3 = Inverse for right argument
| list3 =
*[[Primitive_recursive_function#Predecessor|Predecessor (0)]]
*[[Subtraction|Subtraction (1)]]
*[[Division (mathematics)|Division (2)]]
*[[Logarithm|Logarithm (3)]]
*[[Super-logarithm|Super-logarithm (4)]]
| group4 = Related articles
| list4 =
*[[Ackermann function]]
*[[Conway chained arrow notation]]
*[[Grzegorczyk hierarchy]]
*[[Knuth's up-arrow notation]]
*[[Steinhaus–Moser notation]]
}}

Template:Navbox test

2025-05-22T13:32:16Z

MphLee: Created page with "{{Navbox | name=Navboxtest | title=Hyperoperations | listclass=hlist | group1 = Primary | list1 = *Successor (0) *Addition|Additio..."

{{Navbox
| name=Navboxtest
| title=[[Hyperoperation|Hyperoperations]]
| listclass=hlist
| group1 = Primary
| list1 =
*[[Successor function|Successor (0)]]
*[[Addition|Addition (1)]]
*[[Multiplication|Multiplication (2)]]
*[[Exponentiation|Exponentiation (3)]]
*[[Tetration|Tetration (4)]]
*[[Pentation|Pentation (5)]]
*[[Hexation|Hexation (6)]]
| group2 = [[Inverse function|Inverse]] for left argument
| list2 =
*[[Primitive_recursive_function#Predecessor|Predecessor (0)]]
*[[Subtraction|Subtraction (1)]]
*[[Division (mathematics)|Division (2)]]
*[[nth root|Root extraction (3)]]
*[[Super-root|Super-root (4)]]
| group3 = Inverse for right argument
| list3 =
*[[Primitive_recursive_function#Predecessor|Predecessor (0)]]
*[[Subtraction|Subtraction (1)]]
*[[Division (mathematics)|Division (2)]]
*[[Logarithm|Logarithm (3)]]
*[[Super-logarithm|Super-logarithm (4)]]
| group4 = Related articles
| list4 =
*[[Ackermann function]]
*[[Conway chained arrow notation]]
*[[Grzegorczyk hierarchy]]
*[[Knuth's up-arrow notation]]
*[[Steinhaus–Moser notation]]
}}<noinclude>
{{documentation|content=

[[Category:Arithmetic navigational boxes]]
[[Category:Number templates]]
}}
</noinclude>

Test

2025-05-22T13:31:19Z

MphLee:

Hyperoperations Wiki

2025-05-19T12:37:51Z

MphLee:

The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By keeping track of the definitions, notations, ideas, and bibliographic references, it aims to create a ''cartographic tool'' for the exploration done at the [[Tetrationforum]]. Additionally, by formally re-expressing all modern mathematical concepts from a preparatory viewpoint that is compatible with ''hyperoperative'' inquiry, the wiki could help bridge the historical gap to contemporary mathematics.

[[File:hw.png|250px|center]]

==Idea==
[[File:hw_2.png|thumb|350px|right|The [[hyperoperations]] emerge from the investigation of the nature of [[arithmetical operations]].]]

Organizing the knowledge and folklore surrounding the [[tetration]] function, as well as the broader circle of ideas encompassed by the term "hyperoperation," presents two major challenges:
<ol style="list-style-type:lower-alpha">
<li> most of the work done in this field is extremely heterogeneous in its style, notation, and often lacks reciprocal acknowledgment; </li>
<li> even when some degree of terminological consensus about the mathematical objects inquired is found, the unclear connection to well-known, standard mathematical theories and terminology leaves one with the doubt of reinventing the wheel or the concern of wasting time in fringe, crackpot mathematics. </li>
</ol>
Solving (a) would increase the internal unity of the field of hyperoperations bringing its fundamental problems into clearer view and allowing us to "focus" on them effectively, while solving (b) would provide access powerful and well-established theories and methods for the field.

The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperations]], effectively addressing problem (a). On the other hand, to ensure it is self-contained, the wiki should provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to structures derived from hyperoperations, presenting them from what could be termed a '''''hyperoperational perspective ([[hPOV]]).''''' Dually, this solves problem (b) by situating all the structures within the landscape of modern and contemporary [[mathematics]].

===The hPOV and the concept of iteration===
Just like the concept of a [[ring]], the concept of hyperoperation serves as a potential avenue for exploring the conceptual nature of [[arithmetical operations]] through [[generalization]] and [[abstraction]]. The approach taken by the [[hPOV]] lies on the assumption that some of the relationships between addition and multiplication have a hierarchical nature: some of them are suggestive of an unidirectional increase in complexity and reveal pathways to explore further in those directions, '''''above and beyond''''', i.e. ''ὑπέρ ([[hyper]])'', '''''sums and products'''''. The path suggested is the one of '''''repetition''''', or '''''doing again, once more''''', i.e. ''iterum ([[iteration]])'' that step that got us from addition to multiplication.

[[File:hw_3.png|thumb|600px|center|Intuition about extending the conceptual relationship between sum and product to the whole space of operations.]]

There is a circular relation between the hPOV and the iterative point of view, a kind of duality.
* At first sight, it may seem that iteration supplies the natural ambient theory where to investigate hyperoperations. Once a process linking addition to multiplication is chosen, the study of Hyperoperations seems to be reduced to '''iterating''' that chosen process beyond them.
*Yet also the opposite is true. Iteration emerges naturally just as one of the many possible ways to understand the connection between addition and multiplication.

This reveals the possibility of an intersection of the two points of view by definition of a deeper and fundamental concept, an intersection where the ''iteration of iteration itself'' could happen.

===Useful pages===
[[Special:AllPages]] brings you to a list of all so far described terms.

Hyperoperations Wiki

2025-05-10T10:09:44Z

MphLee:

The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By creating a ''cartographic tool'' for the exploration done at the [[Tetrationforum]], keeping track of of definitions, terms, ideas, and bibliographic references; and by formally re-expressing all modern mathematical concepts from a preparatory viewpoint that is compatible with ''hyperoperative'' inquiry.

[[File:hw.png|250px|center]]

==Idea==
[[File:hw_2.png|thumb|350px|right|The [[hyperoperations]] emerge from the investigation of the nature of [[arithmetical operations]].]]

The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperations]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to structures derived from hyperoperations, presenting them from what could be termed a '''''hyperoperational perspective ([[hPOV]]).'''''

===The hPOV and the concept of iteration===
Just like the concept of a [[ring]], the concept of hyperoperation serves as a potential avenue for exploring the conceptual nature of [[arithmetical operations]] through [[generalization]] and [[abstraction]]. The approach taken by the [[hPOV]] lies on the assumption that some of the relationships between addition and multiplication have a hierarchical nature: some of them are suggestive of an unidirectional increase in complexity and reveal pathways to explore further in those directions, '''''above and beyond''''', i.e. ''ὑπέρ ([[hyper]])'', '''''sums and products'''''. The path suggested is the one of '''''repetition''''', or '''''doing again, once more''''', i.e. ''iterum ([[iteration]])'' that step that got us from addition to multiplication.

[[File:hw_3.png|thumb|600px|center|Intuition about extending the conceptual relationship between sum and product to the whole space of operations.]]

There is a circular relation between the hPOV and the iterative point of view a kind of duality.
* At first sight it may seem that iteration supplies the ambient theory where to investigate hyperoperations. Hyperoperations seems to be reduced to an '''iteration''' of the link between addition and multiplication that goes beyond them.
*Yet also the opposite is true. Iteration emerges naturally just as one of the many possible ways to understand the connection between addition and multiplication.

This reveals the possibility of an intersection of the two points of view by definition of a deeper and fundamental concept, an intersection where the ''iteration of iteration itself'' happens.

===Useful pages===
[[Special:AllPages]] brings you to a list of all so far described terms.

Hyperoperations Wiki

2025-05-10T09:30:03Z

MphLee:

Hyperoperations Wiki

2025-05-10T09:25:50Z

MphLee:

Hyperoperations Wiki

2025-05-10T09:23:22Z

MphLee:

File:Hw 3.png

2025-05-10T09:18:42Z

MphLee:

Hyperoperations Wiki

2025-05-10T07:59:22Z

MphLee:

The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By creating a "cartographic" tool for the exploration done at the [[Tetrationforum]], keeping track of of definitions, terms, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with "hyperoperative" inquiry.

[[File:hw.png|250px|center]]

=Idea=
[[File:hw_2.png|thumb|350px|right|The hyperoperations emerge from the investigation of the nature of [[arithmetical operations]].]]

The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to "hyperoperations-derived" structures, presenting them from what could be termed a "hyperoperational perspective ([[hPOV]])."

Just like the concept of [[ring]], the concept of hyperoperation is just a possible road to investigate the conceptual nature of the [[arithmetical operations]] by means of [[generalization]] and [[abstraction]].

===Useful pages===
[[Special:AllPages]] brings you to a list of all so far described terms.

File:Hw 2.png

2025-05-10T07:41:58Z

MphLee:

File:Hw.png

2025-05-10T07:30:22Z

MphLee:

Hyperoperations Wiki

2025-05-10T06:57:13Z

MphLee:

The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By creating a "cartographic" tool for the exploration done at the [[Tetrationforum]], keeping track of of definitions, terms, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with "hyperoperative" inquiry.

=Idea=

The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to hyperoperative structures, presenting them from what could be termed a "hyperoperational perspective ([[hPOV]])."

===Useful pages===
[[Special:AllPages]] brings you to a list of all so far described terms.

Test

2025-05-10T06:55:29Z

MphLee:

Hyperoperations Wiki

2025-05-10T06:16:26Z

MphLee:

The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By creating a "cartographic" tool for the exploration done at the [[Tetrationforum]], keeping track of of definitions, terms, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with "hyperoperative" inquiry.

=Idea=

The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to hyperoperative structures, presenting them from what could be termed a "hyperoperational perspective ([[hPOV]])."

[[Special:AllPages]] brings you to a list of all so far described terms.

Hyperoperations Wiki

2025-05-10T06:09:21Z

MphLee:

The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By explaining all the terms necessary to understand the discussions.

=Idea=

The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to hyperoperative structures, presenting them from what could be termed a "hyperoperational perspective ([[hPOV]])."

Thus, we have a twofold objective: to create a "cartographic" tool for the exploration done at the [[Tetrationforum]] by keeping track of of definitions, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with "hyperoperative" inquiry.

[[Special:AllPages]] brings you to a list of all so far described terms.

Hyperoperations Wiki

2025-05-09T22:41:15Z

MphLee:

The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to hyperoperative structures, presenting them from what could be termed a "hyperoperational perspective ([[hPOV]])."

Thus, we have a twofold objective: to create a "cartographic" tool for the exploration and cataloging of definitions, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with "hyperoperative" inquiry.

[[Special:AllPages]] brings you to a list of all so far described terms.

Test

2025-05-09T21:26:29Z

MphLee:

This page is for any testing that you want to do.

$x\in\R$ huhu and $z\in\C$

=Section=
Lorem ipsum

==Subsection==
Lorem ipsum

===1. subsubsection===
Lorem ipsum

===2. subsubsection===
Lorem ipsum

====Test====

end

Help:LaTeX

2025-05-09T21:22:43Z

MphLee:

== Inline formulas ==
<pre>$..\sqrt 3 .$</pre>
$..\sqrt 3 .$

== Display style formulas ==
<pre>$$..\sqrt 3.$$</pre>
$$..\sqrt 3.$$

== Latex macros ==
You can use the following abbreviations:
The following list is an excerpt from an initialization file of MathJax and is to be understand, e.g. the first line, as: Use <pre>\C</pre> for $\C$.
<pre>
C: '\\mathbb{C}', /* the complex numbers */
N: '\\mathbb{N}', /* the natural numbers */
Q: '\\mathbb{Q}', /* the rational numbers */
R: '\\mathbb{R}', /* the real numbers */
Z: '\\mathbb{Z}', /* the integer numbers */
ph: '\\varphi',
eps: '\\varepsilon',
th: '\\vartheta',

/* some extre macros for ease of use; these are non-standard! */
F: '\\mathbb{F}', /* a finite field */
HH: '\\mathcal{H}', /* a Hilbert space */
bszero: '\\boldsymbol{0}', /* vector of zeros */
bsone: '\\boldsymbol{1}', /* vector of ones */
bst: '\\boldsymbol{t}', /* a vector 't' */
bsv: '\\boldsymbol{v}', /* a vector 'v' */
bsw: '\\boldsymbol{w}', /* a vector 'w' */
bsx: '\\boldsymbol{x}', /* a vector 'x' */
bsy: '\\boldsymbol{y}', /* a vector 'y' */
bsz: '\\boldsymbol{z}', /* a vector 'z' */
bsDelta: '\\boldsymbol{\\Delta}', /* a vector '\Delta' */
E: '\\mathrm{e}', /* the exponential */
rd: '\\,\\mathrm{d}', /* roman d for use in integrals: $\int f(x) \rd x$ */
rdelta: '\\,\\delta', /* delta operator for use in sums */
rD: '\\mathrm{D}', /* differential operator D */

/* example from MathJax on how to define macros with parameters: */
/* bold: ['{\\bf #1}', 1] */

RR: '\\mathbb{R}',
ZZ: '\\mathbb{Z}',
NN: '\\mathbb{N}',
QQ: '\\mathbb{Q}',
CC: '\\mathbb{C}',
FF: '\\mathbb{F}'
</pre>

Main Page

2025-05-09T21:06:07Z

MphLee: MphLee moved page Main Page to Hyperoperations Wiki

#REDIRECT [[Hyperoperations Wiki]]

Hyperoperations Wiki

2025-05-09T21:06:07Z

MphLee: MphLee moved page Main Page to Hyperoperations Wiki

Hello, Welcome to the Hyperoperations Wiki.

[[Special:AllPages]] brings you to a list of all so far described terms.

Talk:Schröder equation

2025-05-09T19:48:46Z

MphLee: /* test */ new section

Henryk, I corrected the misprint.

Now I think, how about to define new function $G$, let

$$G(z)=\log_c\!\Big(\sigma(z)\Big)$$

? [[User:Kouznetsov|Kouznetsov]] 16:09, 15 December 2012 (CET)

== test ==

000