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"associativity degree" of an operation (from MathSE) - MphLee - 05/19/2014

User Carucell on MathSE posted this awesome question.

http://math.stackexchange.com/questions/788017/superassociative-operation#comment1657341_788017

He asks if exist a binary operation * over the naturals wich has an "infinite degree of associativity".The associativity degree is the number of "hyperoperators" of * that are associative.

The associativity degree is the term I'm going to use to describe better the question.

let define for every its "hyperoperations family" in this way







Lets call the image of this indexed family or in other word we define a function that takes a binary operation and gives the image of its "hyperoperations" indexed family.



The associative degree of is the number of associative binary operations in





The question of the user Carucell is if exist a binary operation with associative degree .
We know that for example




-Another interesting question is if there are operations with associative degree greater than 2 but finite.

-Or maybe another question I have in mind is: every associative operation has at least degree one, do the converse hold? Does every operation with Adegree one is associative?

We can abtain other kind of associative degree functions replacing the construction of the "hyperoperation family" with another definition of "hyperoperation family" in order to obtain different and define a general concept of


RE: "associativity degree" of an operation (from MathSE) - MphLee - 05/21/2014

The user r. e. s. gave an interesting proof about the associativity degree.

http://math.stackexchange.com/questions/788017/superassociative-operation