Binary operations applied to numbers

A binary operation is a calculus that combines two elements to obtain another elements. It seems quite simple for numbers, because we usually imagine it as a simple sum or product. However, also in the case of numbers, a binary operation can be extremely fascinating if we consider it in a generalized form. Here the reader can find several examples of generalized sums for different sets of numbers (Fibonacci, Mersenne, Fermat, q-numbers, repunits and many other numbers). These sets can form groupoid which possess different binary operators. As we will see at the end of this exposition of cases, the most relevant finding is that different integer sequences can have the same binary operator and that, consequently, can be used as different representations of the same groupoid.