Where can I learn more about commutative hyperoperations?












4












$begingroup$


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 especially interested in whether this is a "natural" sequence of definitions.



Note that the expression $k^{log_k(a)+log_k(b)}, k>0$ is independent of $k$, since it always equals $ab$. On the other hand, the expression $k^{log_k(a)log_k(b)}$ is dependent on $k$, and this makes me wonder whether there isn't a "better", more "natural" sequence out there.






reference-request hyperoperation







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    By this criteria, ordinary multiplication isn't natural, because 'constructing' it out of addition requires you to choose which real number will act as the multiplicative unit. (which we will henceforth call '1')
    $endgroup$
    – Hurkyl
    May 12 '13 at 20:25






  • 2




    $begingroup$
    If anyone is interested, it's likely that Albert Bennet's 1915 paper was raised from obscurity when I came across it around 2001 or 2002 and sent Ioannis Galidakis a copy to include in his on-line bibliography on higher order operations, since Bennet's paper was overlooked by Knoebel in Knoebel's 1981 survey paper Exponentials Reiterated.
    $endgroup$
    – Dave L. Renfro
    May 13 '13 at 21:37






  • 1




    $begingroup$
    @DaveL.Renfro, that's pretty cool. Where did you find it?
    $endgroup$
    – goblin
    May 13 '13 at 21:47






  • 1




    $begingroup$
    I came across it while going through journal volumes in a university library. There are quite a few journals (probably over 50) of which during the past 25 years or so I've flipped through every page of every volume, "data mining" interesting (to me) mathematical items.
    $endgroup$
    – Dave L. Renfro
    May 13 '13 at 21:58








  • 1




    $begingroup$
    @MphLee: (and user18921) I've hardly looked at the paper, other than filing it away with a lot of literature I've accumulated on tetration and related topics, literature I sent copies of to Ioannis as I encountered it, at least up until around 2008 or 2009. This started when he expressed a lot of interest in this 9 September 1999 post, which was preceded by this 2 September 1999 sci.math post. Incidentally, one of the things I managed to eventually track down (continued)
    $endgroup$
    – Dave L. Renfro
    May 14 '13 at 14:20
















4












$begingroup$


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 especially interested in whether this is a "natural" sequence of definitions.



Note that the expression $k^{log_k(a)+log_k(b)}, k>0$ is independent of $k$, since it always equals $ab$. On the other hand, the expression $k^{log_k(a)log_k(b)}$ is dependent on $k$, and this makes me wonder whether there isn't a "better", more "natural" sequence out there.






reference-request hyperoperation







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    By this criteria, ordinary multiplication isn't natural, because 'constructing' it out of addition requires you to choose which real number will act as the multiplicative unit. (which we will henceforth call '1')
    $endgroup$
    – Hurkyl
    May 12 '13 at 20:25






  • 2




    $begingroup$
    If anyone is interested, it's likely that Albert Bennet's 1915 paper was raised from obscurity when I came across it around 2001 or 2002 and sent Ioannis Galidakis a copy to include in his on-line bibliography on higher order operations, since Bennet's paper was overlooked by Knoebel in Knoebel's 1981 survey paper Exponentials Reiterated.
    $endgroup$
    – Dave L. Renfro
    May 13 '13 at 21:37






  • 1




    $begingroup$
    @DaveL.Renfro, that's pretty cool. Where did you find it?
    $endgroup$
    – goblin
    May 13 '13 at 21:47






  • 1




    $begingroup$
    I came across it while going through journal volumes in a university library. There are quite a few journals (probably over 50) of which during the past 25 years or so I've flipped through every page of every volume, "data mining" interesting (to me) mathematical items.
    $endgroup$
    – Dave L. Renfro
    May 13 '13 at 21:58








  • 1




    $begingroup$
    @MphLee: (and user18921) I've hardly looked at the paper, other than filing it away with a lot of literature I've accumulated on tetration and related topics, literature I sent copies of to Ioannis as I encountered it, at least up until around 2008 or 2009. This started when he expressed a lot of interest in this 9 September 1999 post, which was preceded by this 2 September 1999 sci.math post. Incidentally, one of the things I managed to eventually track down (continued)
    $endgroup$
    – Dave L. Renfro
    May 14 '13 at 14:20














4












4








4


3



$begingroup$


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 especially interested in whether this is a "natural" sequence of definitions.



Note that the expression $k^{log_k(a)+log_k(b)}, k>0$ is independent of $k$, since it always equals $ab$. On the other hand, the expression $k^{log_k(a)log_k(b)}$ is dependent on $k$, and this makes me wonder whether there isn't a "better", more "natural" sequence out there.






reference-request hyperoperation







share|cite|improve this question











$endgroup$




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 especially interested in whether this is a "natural" sequence of definitions.



Note that the expression $k^{log_k(a)+log_k(b)}, k>0$ is independent of $k$, since it always equals $ab$. On the other hand, the expression $k^{log_k(a)log_k(b)}$ is dependent on $k$, and this makes me wonder whether there isn't a "better", more "natural" sequence out there.






reference-request hyperoperation




reference-request hyperoperation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited May 8 '13 at 16:34







goblin

















asked May 6 '13 at 11:26









goblingoblin

37.1k1159194




37.1k1159194








  • 1




    $begingroup$
    By this criteria, ordinary multiplication isn't natural, because 'constructing' it out of addition requires you to choose which real number will act as the multiplicative unit. (which we will henceforth call '1')
    $endgroup$
    – Hurkyl
    May 12 '13 at 20:25






  • 2




    $begingroup$
    If anyone is interested, it's likely that Albert Bennet's 1915 paper was raised from obscurity when I came across it around 2001 or 2002 and sent Ioannis Galidakis a copy to include in his on-line bibliography on higher order operations, since Bennet's paper was overlooked by Knoebel in Knoebel's 1981 survey paper Exponentials Reiterated.
    $endgroup$
    – Dave L. Renfro
    May 13 '13 at 21:37






  • 1




    $begingroup$
    @DaveL.Renfro, that's pretty cool. Where did you find it?
    $endgroup$
    – goblin
    May 13 '13 at 21:47






  • 1




    $begingroup$
    I came across it while going through journal volumes in a university library. There are quite a few journals (probably over 50) of which during the past 25 years or so I've flipped through every page of every volume, "data mining" interesting (to me) mathematical items.
    $endgroup$
    – Dave L. Renfro
    May 13 '13 at 21:58








  • 1




    $begingroup$
    @MphLee: (and user18921) I've hardly looked at the paper, other than filing it away with a lot of literature I've accumulated on tetration and related topics, literature I sent copies of to Ioannis as I encountered it, at least up until around 2008 or 2009. This started when he expressed a lot of interest in this 9 September 1999 post, which was preceded by this 2 September 1999 sci.math post. Incidentally, one of the things I managed to eventually track down (continued)
    $endgroup$
    – Dave L. Renfro
    May 14 '13 at 14:20














  • 1




    $begingroup$
    By this criteria, ordinary multiplication isn't natural, because 'constructing' it out of addition requires you to choose which real number will act as the multiplicative unit. (which we will henceforth call '1')
    $endgroup$
    – Hurkyl
    May 12 '13 at 20:25






  • 2




    $begingroup$
    If anyone is interested, it's likely that Albert Bennet's 1915 paper was raised from obscurity when I came across it around 2001 or 2002 and sent Ioannis Galidakis a copy to include in his on-line bibliography on higher order operations, since Bennet's paper was overlooked by Knoebel in Knoebel's 1981 survey paper Exponentials Reiterated.
    $endgroup$
    – Dave L. Renfro
    May 13 '13 at 21:37






  • 1




    $begingroup$
    @DaveL.Renfro, that's pretty cool. Where did you find it?
    $endgroup$
    – goblin
    May 13 '13 at 21:47






  • 1




    $begingroup$
    I came across it while going through journal volumes in a university library. There are quite a few journals (probably over 50) of which during the past 25 years or so I've flipped through every page of every volume, "data mining" interesting (to me) mathematical items.
    $endgroup$
    – Dave L. Renfro
    May 13 '13 at 21:58








  • 1




    $begingroup$
    @MphLee: (and user18921) I've hardly looked at the paper, other than filing it away with a lot of literature I've accumulated on tetration and related topics, literature I sent copies of to Ioannis as I encountered it, at least up until around 2008 or 2009. This started when he expressed a lot of interest in this 9 September 1999 post, which was preceded by this 2 September 1999 sci.math post. Incidentally, one of the things I managed to eventually track down (continued)
    $endgroup$
    – Dave L. Renfro
    May 14 '13 at 14:20








1




1




$begingroup$
By this criteria, ordinary multiplication isn't natural, because 'constructing' it out of addition requires you to choose which real number will act as the multiplicative unit. (which we will henceforth call '1')
$endgroup$
– Hurkyl
May 12 '13 at 20:25




$begingroup$
By this criteria, ordinary multiplication isn't natural, because 'constructing' it out of addition requires you to choose which real number will act as the multiplicative unit. (which we will henceforth call '1')
$endgroup$
– Hurkyl
May 12 '13 at 20:25




2




2




$begingroup$
If anyone is interested, it's likely that Albert Bennet's 1915 paper was raised from obscurity when I came across it around 2001 or 2002 and sent Ioannis Galidakis a copy to include in his on-line bibliography on higher order operations, since Bennet's paper was overlooked by Knoebel in Knoebel's 1981 survey paper Exponentials Reiterated.
$endgroup$
– Dave L. Renfro
May 13 '13 at 21:37




$begingroup$
If anyone is interested, it's likely that Albert Bennet's 1915 paper was raised from obscurity when I came across it around 2001 or 2002 and sent Ioannis Galidakis a copy to include in his on-line bibliography on higher order operations, since Bennet's paper was overlooked by Knoebel in Knoebel's 1981 survey paper Exponentials Reiterated.
$endgroup$
– Dave L. Renfro
May 13 '13 at 21:37




1




1




$begingroup$
@DaveL.Renfro, that's pretty cool. Where did you find it?
$endgroup$
– goblin
May 13 '13 at 21:47




$begingroup$
@DaveL.Renfro, that's pretty cool. Where did you find it?
$endgroup$
– goblin
May 13 '13 at 21:47




1




1




$begingroup$
I came across it while going through journal volumes in a university library. There are quite a few journals (probably over 50) of which during the past 25 years or so I've flipped through every page of every volume, "data mining" interesting (to me) mathematical items.
$endgroup$
– Dave L. Renfro
May 13 '13 at 21:58






$begingroup$
I came across it while going through journal volumes in a university library. There are quite a few journals (probably over 50) of which during the past 25 years or so I've flipped through every page of every volume, "data mining" interesting (to me) mathematical items.
$endgroup$
– Dave L. Renfro
May 13 '13 at 21:58






1




1




$begingroup$
@MphLee: (and user18921) I've hardly looked at the paper, other than filing it away with a lot of literature I've accumulated on tetration and related topics, literature I sent copies of to Ioannis as I encountered it, at least up until around 2008 or 2009. This started when he expressed a lot of interest in this 9 September 1999 post, which was preceded by this 2 September 1999 sci.math post. Incidentally, one of the things I managed to eventually track down (continued)
$endgroup$
– Dave L. Renfro
May 14 '13 at 14:20




$begingroup$
@MphLee: (and user18921) I've hardly looked at the paper, other than filing it away with a lot of literature I've accumulated on tetration and related topics, literature I sent copies of to Ioannis as I encountered it, at least up until around 2008 or 2009. This started when he expressed a lot of interest in this 9 September 1999 post, which was preceded by this 2 September 1999 sci.math post. Incidentally, one of the things I managed to eventually track down (continued)
$endgroup$
– Dave L. Renfro
May 14 '13 at 14:20










2 Answers
2






active

oldest

votes


















4





+100







$begingroup$

Try to check the chapter $1$ (from Pag $9$) of this book (New Mathematical Objects-C.A.Rubtsov) :



Here the autor creates a kind of generalization of the Albert Bennet's Hyperoperations. But what he does is much more general. He creates a procedure that he calls $ω$-reflection using a function of connection $f$.



In the book the only function he uses is $f(x):=k^x$ where $kgt 1$ is called in the book "factor of image".



Then he defines an infinite hierarchy of "reflexive binary operations" that are homomorphic via $f$:



$xcirc_iy=x+y$ if $i=1$



$f(x)circ_{i+1}f(y)=f(xcirc_iy)$



and we have that



$xcirc_{i+1}y=f(f^{circ-1}(x)circ_if^{circ-1}(y))$



The first chapter the autor puts more attention on the homomorphic operation $circ_{3}$ that is denoted by $odot$ in the book and called "reflexive multiplication". Since he uses the exponentiation as function of connection, $circ_{3}$ is isomophic to the mutiplication, then commutative and associative.



$k^x odot k^y = k^{a times b}$



In this chapter the autor builds an infinite numbers of what he calls reflexive functions, reflexive (homomorphic) binary operations , reflexive algebras (that are homomorphics via f) and other mathematical objects in this way:



let $F$ a bijection $F:Bbb R rightarrow Bbb R$ (function of connection)



he calls $f'$ the reflexive image of the function of $f$ via $F$ if we have



$f'circ F=F circ f$



If you want to go deeper, on Tetration forum there is a topic where the possible propeties/evaluation of reflexive binary operations with non-integer indexes $ile 2$ using tetration is discussed: Rational Operators




More informations:




$1$ The first autor is C.A.Rubtsov that with G.F.Romerio worked on the
Hyperoperations topic:



(Ackermann's function and new arithematical operations-Rubtsov,
Romerio-2004)



$2$ About Albert Bennet here a pdf about his commutative Hyperoperations:



(Note on an Operation of the Thrid Grade- Albert A.Bennet )





if the link is broken try these



http://numbers.newmail.ru/pdf/book_eng.pdf



http://numbers.newmail.ru/english/09.htm





share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    sounds interesting, but that link at the top of your posts goes to a "Error 404. Document not found" page.
    $endgroup$
    – goblin
    May 13 '13 at 21:49










  • $begingroup$
    @user18921 updated the link, anyways if doesn not work, try several times to open it, or copy and paste the http.
    $endgroup$
    – MphLee
    May 14 '13 at 7:36










  • $begingroup$
    @user18921 Now can you see the book?(in the second link you must chose "BOOK_ENG.PDF" )
    $endgroup$
    – MphLee
    May 14 '13 at 8:13








  • 1




    $begingroup$
    Okay, this link worked. Thank you.
    $endgroup$
    – goblin
    May 14 '13 at 11:49



















1












$begingroup$

This answer outlines an application for these operators: defining an infinite sequence of abelian group structures with an infinite sequence of triplets like (+, -, 0), (×, ÷, 1), etc.






share|cite|improve this answer









$endgroup$














    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f383168%2fwhere-can-i-learn-more-about-commutative-hyperoperations%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4





    +100







    $begingroup$

    Try to check the chapter $1$ (from Pag $9$) of this book (New Mathematical Objects-C.A.Rubtsov) :



    Here the autor creates a kind of generalization of the Albert Bennet's Hyperoperations. But what he does is much more general. He creates a procedure that he calls $ω$-reflection using a function of connection $f$.



    In the book the only function he uses is $f(x):=k^x$ where $kgt 1$ is called in the book "factor of image".



    Then he defines an infinite hierarchy of "reflexive binary operations" that are homomorphic via $f$:



    $xcirc_iy=x+y$ if $i=1$



    $f(x)circ_{i+1}f(y)=f(xcirc_iy)$



    and we have that



    $xcirc_{i+1}y=f(f^{circ-1}(x)circ_if^{circ-1}(y))$



    The first chapter the autor puts more attention on the homomorphic operation $circ_{3}$ that is denoted by $odot$ in the book and called "reflexive multiplication". Since he uses the exponentiation as function of connection, $circ_{3}$ is isomophic to the mutiplication, then commutative and associative.



    $k^x odot k^y = k^{a times b}$



    In this chapter the autor builds an infinite numbers of what he calls reflexive functions, reflexive (homomorphic) binary operations , reflexive algebras (that are homomorphics via f) and other mathematical objects in this way:



    let $F$ a bijection $F:Bbb R rightarrow Bbb R$ (function of connection)



    he calls $f'$ the reflexive image of the function of $f$ via $F$ if we have



    $f'circ F=F circ f$



    If you want to go deeper, on Tetration forum there is a topic where the possible propeties/evaluation of reflexive binary operations with non-integer indexes $ile 2$ using tetration is discussed: Rational Operators




    More informations:




    $1$ The first autor is C.A.Rubtsov that with G.F.Romerio worked on the
    Hyperoperations topic:



    (Ackermann's function and new arithematical operations-Rubtsov,
    Romerio-2004)



    $2$ About Albert Bennet here a pdf about his commutative Hyperoperations:



    (Note on an Operation of the Thrid Grade- Albert A.Bennet )





    if the link is broken try these



    http://numbers.newmail.ru/pdf/book_eng.pdf
    

    
http://numbers.newmail.ru/english/09.htm





    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      sounds interesting, but that link at the top of your posts goes to a "Error 404. Document not found" page.
      $endgroup$
      – goblin
      May 13 '13 at 21:49










    • $begingroup$
      @user18921 updated the link, anyways if doesn not work, try several times to open it, or copy and paste the http.
      $endgroup$
      – MphLee
      May 14 '13 at 7:36










    • $begingroup$
      @user18921 Now can you see the book?(in the second link you must chose "BOOK_ENG.PDF" )
      $endgroup$
      – MphLee
      May 14 '13 at 8:13








    • 1




      $begingroup$
      Okay, this link worked. Thank you.
      $endgroup$
      – goblin
      May 14 '13 at 11:49
















    4





    +100







    $begingroup$

    Try to check the chapter $1$ (from Pag $9$) of this book (New Mathematical Objects-C.A.Rubtsov) :



    Here the autor creates a kind of generalization of the Albert Bennet's Hyperoperations. But what he does is much more general. He creates a procedure that he calls $ω$-reflection using a function of connection $f$.



    In the book the only function he uses is $f(x):=k^x$ where $kgt 1$ is called in the book "factor of image".



    Then he defines an infinite hierarchy of "reflexive binary operations" that are homomorphic via $f$:



    $xcirc_iy=x+y$ if $i=1$



    $f(x)circ_{i+1}f(y)=f(xcirc_iy)$



    and we have that



    $xcirc_{i+1}y=f(f^{circ-1}(x)circ_if^{circ-1}(y))$



    The first chapter the autor puts more attention on the homomorphic operation $circ_{3}$ that is denoted by $odot$ in the book and called "reflexive multiplication". Since he uses the exponentiation as function of connection, $circ_{3}$ is isomophic to the mutiplication, then commutative and associative.



    $k^x odot k^y = k^{a times b}$



    In this chapter the autor builds an infinite numbers of what he calls reflexive functions, reflexive (homomorphic) binary operations , reflexive algebras (that are homomorphics via f) and other mathematical objects in this way:



    let $F$ a bijection $F:Bbb R rightarrow Bbb R$ (function of connection)



    he calls $f'$ the reflexive image of the function of $f$ via $F$ if we have



    $f'circ F=F circ f$



    If you want to go deeper, on Tetration forum there is a topic where the possible propeties/evaluation of reflexive binary operations with non-integer indexes $ile 2$ using tetration is discussed: Rational Operators




    More informations:




    $1$ The first autor is C.A.Rubtsov that with G.F.Romerio worked on the
    Hyperoperations topic:



    (Ackermann's function and new arithematical operations-Rubtsov,
    Romerio-2004)



    $2$ About Albert Bennet here a pdf about his commutative Hyperoperations:



    (Note on an Operation of the Thrid Grade- Albert A.Bennet )





    if the link is broken try these



    http://numbers.newmail.ru/pdf/book_eng.pdf
    

    
http://numbers.newmail.ru/english/09.htm





    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      sounds interesting, but that link at the top of your posts goes to a "Error 404. Document not found" page.
      $endgroup$
      – goblin
      May 13 '13 at 21:49










    • $begingroup$
      @user18921 updated the link, anyways if doesn not work, try several times to open it, or copy and paste the http.
      $endgroup$
      – MphLee
      May 14 '13 at 7:36










    • $begingroup$
      @user18921 Now can you see the book?(in the second link you must chose "BOOK_ENG.PDF" )
      $endgroup$
      – MphLee
      May 14 '13 at 8:13








    • 1




      $begingroup$
      Okay, this link worked. Thank you.
      $endgroup$
      – goblin
      May 14 '13 at 11:49














    4





    +100







    4





    +100



    4




    +100



    $begingroup$

    Try to check the chapter $1$ (from Pag $9$) of this book (New Mathematical Objects-C.A.Rubtsov) :



    Here the autor creates a kind of generalization of the Albert Bennet's Hyperoperations. But what he does is much more general. He creates a procedure that he calls $ω$-reflection using a function of connection $f$.



    In the book the only function he uses is $f(x):=k^x$ where $kgt 1$ is called in the book "factor of image".



    Then he defines an infinite hierarchy of "reflexive binary operations" that are homomorphic via $f$:



    $xcirc_iy=x+y$ if $i=1$



    $f(x)circ_{i+1}f(y)=f(xcirc_iy)$



    and we have that



    $xcirc_{i+1}y=f(f^{circ-1}(x)circ_if^{circ-1}(y))$



    The first chapter the autor puts more attention on the homomorphic operation $circ_{3}$ that is denoted by $odot$ in the book and called "reflexive multiplication". Since he uses the exponentiation as function of connection, $circ_{3}$ is isomophic to the mutiplication, then commutative and associative.



    $k^x odot k^y = k^{a times b}$



    In this chapter the autor builds an infinite numbers of what he calls reflexive functions, reflexive (homomorphic) binary operations , reflexive algebras (that are homomorphics via f) and other mathematical objects in this way:



    let $F$ a bijection $F:Bbb R rightarrow Bbb R$ (function of connection)



    he calls $f'$ the reflexive image of the function of $f$ via $F$ if we have



    $f'circ F=F circ f$



    If you want to go deeper, on Tetration forum there is a topic where the possible propeties/evaluation of reflexive binary operations with non-integer indexes $ile 2$ using tetration is discussed: Rational Operators




    More informations:




    $1$ The first autor is C.A.Rubtsov that with G.F.Romerio worked on the
    Hyperoperations topic:



    (Ackermann's function and new arithematical operations-Rubtsov,
    Romerio-2004)



    $2$ About Albert Bennet here a pdf about his commutative Hyperoperations:



    (Note on an Operation of the Thrid Grade- Albert A.Bennet )





    if the link is broken try these



    http://numbers.newmail.ru/pdf/book_eng.pdf
    

    
http://numbers.newmail.ru/english/09.htm





    share|cite|improve this answer











    $endgroup$



    Try to check the chapter $1$ (from Pag $9$) of this book (New Mathematical Objects-C.A.Rubtsov) :



    Here the autor creates a kind of generalization of the Albert Bennet's Hyperoperations. But what he does is much more general. He creates a procedure that he calls $ω$-reflection using a function of connection $f$.



    In the book the only function he uses is $f(x):=k^x$ where $kgt 1$ is called in the book "factor of image".



    Then he defines an infinite hierarchy of "reflexive binary operations" that are homomorphic via $f$:



    $xcirc_iy=x+y$ if $i=1$



    $f(x)circ_{i+1}f(y)=f(xcirc_iy)$



    and we have that



    $xcirc_{i+1}y=f(f^{circ-1}(x)circ_if^{circ-1}(y))$



    The first chapter the autor puts more attention on the homomorphic operation $circ_{3}$ that is denoted by $odot$ in the book and called "reflexive multiplication". Since he uses the exponentiation as function of connection, $circ_{3}$ is isomophic to the mutiplication, then commutative and associative.



    $k^x odot k^y = k^{a times b}$



    In this chapter the autor builds an infinite numbers of what he calls reflexive functions, reflexive (homomorphic) binary operations , reflexive algebras (that are homomorphics via f) and other mathematical objects in this way:



    let $F$ a bijection $F:Bbb R rightarrow Bbb R$ (function of connection)



    he calls $f'$ the reflexive image of the function of $f$ via $F$ if we have



    $f'circ F=F circ f$



    If you want to go deeper, on Tetration forum there is a topic where the possible propeties/evaluation of reflexive binary operations with non-integer indexes $ile 2$ using tetration is discussed: Rational Operators




    More informations:




    $1$ The first autor is C.A.Rubtsov that with G.F.Romerio worked on the
    Hyperoperations topic:



    (Ackermann's function and new arithematical operations-Rubtsov,
    Romerio-2004)



    $2$ About Albert Bennet here a pdf about his commutative Hyperoperations:



    (Note on an Operation of the Thrid Grade- Albert A.Bennet )





    if the link is broken try these



    http://numbers.newmail.ru/pdf/book_eng.pdf
    

    
http://numbers.newmail.ru/english/09.htm






    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited May 23 '15 at 15:15

























    answered May 12 '13 at 17:13









    MphLeeMphLee

    1,17711237




    1,17711237








    • 1




      $begingroup$
      sounds interesting, but that link at the top of your posts goes to a "Error 404. Document not found" page.
      $endgroup$
      – goblin
      May 13 '13 at 21:49










    • $begingroup$
      @user18921 updated the link, anyways if doesn not work, try several times to open it, or copy and paste the http.
      $endgroup$
      – MphLee
      May 14 '13 at 7:36










    • $begingroup$
      @user18921 Now can you see the book?(in the second link you must chose "BOOK_ENG.PDF" )
      $endgroup$
      – MphLee
      May 14 '13 at 8:13








    • 1




      $begingroup$
      Okay, this link worked. Thank you.
      $endgroup$
      – goblin
      May 14 '13 at 11:49














    • 1




      $begingroup$
      sounds interesting, but that link at the top of your posts goes to a "Error 404. Document not found" page.
      $endgroup$
      – goblin
      May 13 '13 at 21:49










    • $begingroup$
      @user18921 updated the link, anyways if doesn not work, try several times to open it, or copy and paste the http.
      $endgroup$
      – MphLee
      May 14 '13 at 7:36










    • $begingroup$
      @user18921 Now can you see the book?(in the second link you must chose "BOOK_ENG.PDF" )
      $endgroup$
      – MphLee
      May 14 '13 at 8:13








    • 1




      $begingroup$
      Okay, this link worked. Thank you.
      $endgroup$
      – goblin
      May 14 '13 at 11:49








    1




    1




    $begingroup$
    sounds interesting, but that link at the top of your posts goes to a "Error 404. Document not found" page.
    $endgroup$
    – goblin
    May 13 '13 at 21:49




    $begingroup$
    sounds interesting, but that link at the top of your posts goes to a "Error 404. Document not found" page.
    $endgroup$
    – goblin
    May 13 '13 at 21:49












    $begingroup$
    @user18921 updated the link, anyways if doesn not work, try several times to open it, or copy and paste the http.
    $endgroup$
    – MphLee
    May 14 '13 at 7:36




    $begingroup$
    @user18921 updated the link, anyways if doesn not work, try several times to open it, or copy and paste the http.
    $endgroup$
    – MphLee
    May 14 '13 at 7:36












    $begingroup$
    @user18921 Now can you see the book?(in the second link you must chose "BOOK_ENG.PDF" )
    $endgroup$
    – MphLee
    May 14 '13 at 8:13






    $begingroup$
    @user18921 Now can you see the book?(in the second link you must chose "BOOK_ENG.PDF" )
    $endgroup$
    – MphLee
    May 14 '13 at 8:13






    1




    1




    $begingroup$
    Okay, this link worked. Thank you.
    $endgroup$
    – goblin
    May 14 '13 at 11:49




    $begingroup$
    Okay, this link worked. Thank you.
    $endgroup$
    – goblin
    May 14 '13 at 11:49











    1












    $begingroup$

    This answer outlines an application for these operators: defining an infinite sequence of abelian group structures with an infinite sequence of triplets like (+, -, 0), (×, ÷, 1), etc.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      This answer outlines an application for these operators: defining an infinite sequence of abelian group structures with an infinite sequence of triplets like (+, -, 0), (×, ÷, 1), etc.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        This answer outlines an application for these operators: defining an infinite sequence of abelian group structures with an infinite sequence of triplets like (+, -, 0), (×, ÷, 1), etc.






        share|cite|improve this answer









        $endgroup$



        This answer outlines an application for these operators: defining an infinite sequence of abelian group structures with an infinite sequence of triplets like (+, -, 0), (×, ÷, 1), etc.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 5 at 15:36









        ismaelismael

        281216




        281216






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f383168%2fwhere-can-i-learn-more-about-commutative-hyperoperations%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Wiesbaden

            Image
            Wappen Deutschlandkarte 50.0825 8.24 117 Koordinaten: 50° 5′  N , 8° 14′  O Basisdaten Bundesland: Hessen Regierungsbezirk: Darmstadt Höhe: 117 m ü. NHN Fläche: 203,93 km 2 Einwohner: 278.654 (31. Dez. 2017) [1] Bevölkerungsdichte: 1366 Einwohner je km 2 Postleitzahlen: 65183–65207, 55246, 55252 Vorlage:Infobox Gemeinde in Deutschland/Wartung/PLZ enthält Text Vorwahlen: 0611, 06122, 06127, 06134 Kfz-Kennzeichen: WI Gemeindeschlüssel: 06 4 14 000 LOCODE: DE WIB NUTS: DE714 Stadtgliederung: 26 Ortsbezirke Adresse der Stadtverwaltung: Schlossplatz 6 65183 Wiesbaden Webpräsenz: www.wiesbaden.de Oberbürgermeister: Sven Gerich (SPD) Lage der Landeshauptstadt Wiesbaden in Hessen und im Regierungsbezirk Darmstadt Offizielles Logo der Landeshauptstadt Wiesbaden Flagge der Landeshauptstadt Wiesbaden Wiesbaden is
            Read more

            Marschland

            Image
            Kilometerweite Marsch in den Vier- und Marschlanden am Elbdeich Die Wedeler Marschlandschaft in Schleswig-Holstein Der Hadelner Kanal zur Entwässerung der Marsch Die tief gelegene Marsch und ein Entwässerungskanal Das Schöpfwerk Otterndorf im Land Hadeln zur Entwässerung der Medem verfügte damals bereits über die größte Kreiselpumpe Europas. Das Schöpfwerk in Neuhaus für die Aue und des Neuhaus-Bülkauer Kanals Als Marsch(land) (v. niederdt., altsächs. mersc ) – auch Masch , Mersch oder Schwemmland genannt – bezeichnet man eine nacheiszeitlich entstandene geomorphologische Landform im Gebiet der nordwestdeutschen Küsten und Flüsse sowie vergleichbare Landformen weltweit. Inhaltsverzeichnis 1 Landschaft 2 Entstehung 3 Marschböden 4 Nutzung 5 Marschgebiete in Deutschland 6 Siehe auch 7 Literatur 8 Weblinks 9 Einzelnachweise Landschaft | Marschen sind generell flache Landstriche ohne natürliche Erhebungen. Si
            Read more

            Dieringhausen

            Image
            Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
            Read more