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I was curious as multiplication is really just a shorthand addition, so whats so special about it? Could we generalise to all hyper operations? Does there exist algebraic structures with these operations?

Joshua Farrell
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    Can you explain how $\sqrt{3}\cdot\sqrt{2}=\sqrt{6}$ is just shorthand addition? – saulspatz Oct 12 '18 at 13:41
  • For integers, it certainly is syntactic sugar for addition, but for rationals and reals, it's a being of its own. – Rushabh Mehta Oct 12 '18 at 13:42
  • Sure, you can create a group where the binary operator is a hyper operator. Groups do not require a specific binary operator. Convention is to use standard addition and multiplication. But this is not required. For example, when you are working with algebraic topology, the homotopy group involves addition of paths which is in a sense an operation of concatenation rather than addition as we normally think of it. – SlipEternal Oct 12 '18 at 13:48
  • For example, do you want some list of properties of $+, \times, \wedge$ that could be used to define some algebraic object with three operations? – GEdgar Oct 13 '18 at 12:47

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The answer is no. Hyperoperations are defined on $\mathbb{N}$ but are not defined on arbitrary semigroups, groups or rings. In a ring, the multiplication has no reason to be defined in terms of the addition (think of the product of two matrices, for instance).

J.-E. Pin
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  • Just curious: is there a term that classifies rings where multiplication and addition do have this hyperoperation relationship? e.g. where 2 * 3 = 2 + 2 + 2? or does this idea have no relevance at all in the context of rings? – Matt D Feb 14 '20 at 21:03
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    @matt-d I am not sure to understand your question. If you define $2$ as $1 + 1$ and $3$ as $1 + 1 + 1$, then the formula $2 * 3 = 2 + 2 + 2$ holds in any ring. The problem is that, in general, the elements of a ring $R$ that are sums of $1$ just form a subring of $R$, isomorphic to $\Bbb Z$ or to $\Bbb Z/{n\Bbb Z}$, depending on the characteristic of $R$. This subring is equal to $R$ if and only if $R$ is (isomorphic to) a quotient of $\Bbb Z$. – J.-E. Pin Feb 14 '20 at 23:08
  • Sorry, I think I'm just missing something obvious... let's say instead of 1, 2, and 3 we name the elements a, b, and c. If all we know is that b = a + a and c = a + a + a, how do we derive the result that b * c = b + b + b? How do the first two statements about the additive operation inform us about the behavior of the multiplicative operation? Is there a relationship between the operations beyond the distributive axiom? (I think this last question might be where I'm confused.) – Matt D Feb 15 '20 at 04:49
  • Thinking about this further: maybe what I'm missing is that "a" is the multiplicative identity in that example, and that fact is what allows us to figure out what b * c equals... is that right? – Matt D Feb 15 '20 at 05:00
  • nvm, I think I found what I was looking for here: https://math.stackexchange.com/questions/2691665/in-what-algebraic-structure-does-repeated-addition-equal-multiplication... appreciate the reply, though! – Matt D Feb 15 '20 at 05:20