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The algebraic structures semi group, monoid and group are defined on a set using only one binary operation.

The algebraic structures ring and field are defined on a set using two binary operations.

My question is: Why did the mathematicians not define an algebraic structure using more than $2$ binary operators?

A probable reasoning can be : modern algebra originated while looking at some generalized version of set of integers. There are $2$ basic operations namely addition and multiplication (subtraction and division are inverse operations respectively) in integers, so those algebraic structures were defined which used $2$ binary operations.

My point is that there are more operations on integers such as factorial(double, triple... factorial as well) for which it might be useful to have an algebraic structure with more than $2$ binary operations.

For the argument that factorial is just the repeated multiplication so ring/field should be sufficient, here is my cross question 'why did we define an algebraic structure with $2$ binary operation while multiplication is just the repeated addition?'

SARTHAK GUPTA
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    Factorial is a unary, not a binary, operation – J. W. Tanner Jan 01 '20 at 21:40
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    Take a look https://en.wikipedia.org/wiki/Composition_ring – Matheus Nunes Jan 01 '20 at 21:42
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    There's no rule against studying structures in larger languages, and sometimes they are important; however, an at-first-glance narrow range of structures (groups, rings, fields, etc.) has turned out through experience to have special importance. Why that is is of course a very hard question ... – Noah Schweber Jan 01 '20 at 21:43
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    (Incidentally, here's another example of a kind of larger-language structure that mathematicians actively care about.) – Noah Schweber Jan 01 '20 at 21:44
  • @J.W.Tanner Sorry, my bad. But there are few operations which require 3 elements from the set. Have a look https://en.wikipedia.org/wiki/Ternary_operation – SARTHAK GUPTA Jan 01 '20 at 21:49
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    We do. The algebra of polynomials over a ring and inner product spaces are two examples of mathematical structures that have three binary operations that all play together in a specific way. Even so, the pairs of operations often work together according to one of the two-operation structures, so it is important to study these basic structures for all they're worth. –  Jan 01 '20 at 21:57
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    Another binary operation is exponentiation: $a^b$ – J. W. Tanner Jan 01 '20 at 22:04
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    All n-ary operations are compositions of binary operations, an old result of Sierpinski (alas probably most won't see this comment since every comment above has been upvoted, so it's now hidden by default). – Bill Dubuque Jan 01 '20 at 22:45
  • Hopf algebras also provide an important class of algebraic structures that are current topics of research. – Eric Canton Jan 02 '20 at 19:07
  • Towards your cross-question at the end of your post: not every multiplication in every ring arises from repeated addition, this is actually fairly special to the integers and similar rings. For example, the polynomial $xy$ is not equal to $(y + \cdots + y)$ for any number of $y$ terms. It's also not really true that $\pi x \in \mathbb{R}[x]$ (here $\pi$ is the ratio of circumference to diameter of a circle) can be made using repeated addition of $x$, though if one allows fractional quanitites of $x$ and also taking limits, we could get around this (e.g. $x + x + x + 0.1x + 0.04x + \dots$). – Eric Canton Jan 02 '20 at 19:14

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