Wikipedia:Auto magma object

In mathematics, a magma object, can be defined in any category $$\mathbf{C}$$ equipped with a distinguished bifunctor $$\otimes : \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C}$$. Since Mag, the category of magmas, has cartesian products, we can therefore consider magma objects in the category Mag. These are called auto magma objects. There is a more direct definition: an auto magma object is a set $$X$$ together with a pair of binary operations $$f,g:X\times X \rightarrow X$$ satisfying $$g(f(x,y),f(x',y')) = f(g(x,x'),g(y,y'))$$ for all $$x,x',y,y'$$ in $$X$$. A medial magma is the special case where these operations are equal.