Generalization of Operations

• A discussion with T.A.

• Generalization of operations such as addition, multiplication, exponentiation, etc. in arithmetic

  • It already exists as the Ackermann function.

• From there, I wonder if we can define operations before addition (n<1).

  • Similar to extending the power function to exponential function.
    • The exponential function extends the power function to real numbers.

• It seems good to find some broad laws and then narrow them down.

  • First, as an intuition, ack(a,b,n) > ack(a,b,n+1).
    • Multiplying 3 by 3 is bigger than adding 3 and 3, right?
    • I want to prove this.
    • By establishing such laws as a basis, we can extend the definition.

• https://math.stackexchange.com/questions/35598/why-are-addition-and-multiplication-commutative-but-not-exponentiation

  • In the case of a^b, the roles of a and b are completely different, so it’s not commutative, right?
    • I feel like I had already come to this conclusion myself.

• A three-dimensional graph as f(a,b)=c

  • Addition

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  • Multiplication

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  • Exponentiation

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  • When the commutative property holds, a graph that is symmetric about the diagonal of the xy plane is formed.