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 the distributive property MphLee Fellow Posts: 184 Threads: 19 Joined: May 2013 05/27/2014, 08:22 PM (05/27/2014, 07:45 PM)andydude Wrote: @MphLee Hyperoperations, in the general sense, are any sequence of binary operations that includes addition and multiplication. The commutative hyperoperations satisfy this property because $\exp^0(\ln^0(a) + \ln^0(b)) = a + b$ and $\exp^1(\ln^1(a) + \ln^1(b)) = e^{\ln(a) + \ln(b)} = e^{\ln(a)}e^{\ln(b)} = a \times b$. That formula is the starting point, it is the definition of commutative hyperoperations. The fact that it contains addition and multiplication can be discussed and proved from the definition. I'm aware of this, of the Bennet's definition and I'm aware of the Rubtsov's generalizzation using different bases values (reflexive operations sequence). I'm even aware that the term Hyperoperations usually means (can be formalized as) an indexed family of binary operations $\{*_i\}_{i \in I}$ whith addition, multiplication and exponentiation belonging to the image of the indexed family (the image of the family is defined to be the image of the set of indexes- set of ranks- via the indicization function). This definition is the one I found on Wikipedia and is very smart even if it cuts the Commutative hyperoperations out of the game (Maybe we can make a weaker concept of Hyperoperations Family without the exponentiation requirement, I would call them Weak Hyperoperations Families)... Anyways I'm very courious...I was not able to find references about this terminology and I did not even find who introduced this formal definition. MathStackExchange account:MphLee Fundamental Law $(\sigma+1)0=\sigma (\sigma+1)$ « Next Oldest | Next Newest »

 Messages In This Thread the distributive property - by tommy1729 - 09/25/2010, 12:05 AM RE: the distributive property - by tommy1729 - 09/28/2010, 12:22 PM RE: the distributive property - by tommy1729 - 10/13/2010, 11:07 PM RE: the distributive property - by tommy1729 - 05/23/2014, 11:12 PM RE: the distributive property - by MphLee - 05/24/2014, 09:46 AM RE: the distributive property - by tommy1729 - 05/25/2014, 12:16 AM RE: the distributive property - by MphLee - 05/25/2014, 09:35 AM RE: the distributive property - by tommy1729 - 05/26/2014, 11:37 PM RE: the distributive property - by MphLee - 05/27/2014, 07:43 AM RE: the distributive property - by tommy1729 - 05/27/2014, 12:23 PM RE: the distributive property - by MphLee - 05/27/2014, 12:46 PM RE: the distributive property - by tommy1729 - 05/27/2014, 10:49 PM RE: the distributive property - by andydude - 05/27/2014, 07:45 PM RE: the distributive property - by MphLee - 05/27/2014, 08:22 PM RE: the distributive property - by tommy1729 - 05/27/2014, 10:55 PM RE: the distributive property - by MphLee - 06/06/2014, 09:56 AM RE: the distributive property - by tommy1729 - 06/06/2014, 11:56 AM RE: the distributive property - by MphLee - 06/06/2014, 12:09 PM

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