how a new operator is born

For example, there are multiple operators like +, -, * , / etc in math.

I actually wanted to know how a new operator is born.

How the operator is verified? I mean when can we say that, yes this operators belong to X branch of mathematics (ex. arithmetic, algebra etc.).

How can we proof that these collection of operations are enough to deal with X branch of mathematics (ex. arithmetic, algebra etc.).

I think there are research papers that do that. I would like to have reference to some of those papers.

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from math https://ift.tt/3f7Xtzm https://ift.tt/eA8V8J