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Rarely, one uses a letter, such as $f \colon S \times S \rightarrow S$, to denote a binary operation, and traditional notation $f(s_1, s_2)$ for the output of the function $f$ after the input $(s_1, s_2)$.
Most commonly an operation symbol, such as "$\circ$", "$+$", or "$\cdot$" is used to denote the function from $S \times S$ to $S$, and, for example, $s_1 + s_2$ is used to denote the output of the function "$+$" after the input $(s_1, s_2)$.
The following statements and definitions logically rely on the material of this page: Def/Commutative, Def/Commute, Def/Diophantine equation, Def/Distributive, Def/Group, Def/Monoid, Def/Ring, and Def/Vector space
To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Operations