# Auto magma object

In mathematics, a magma object, can be defined in any category $\displaystyle{ \mathbf{C} }$ equipped with a distinguished bifunctor $\displaystyle{ \otimes : \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C} }$. 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 $\displaystyle{ X }$ together with a pair of binary operations $\displaystyle{ f,g:X\times X \rightarrow X }$ satisfying $\displaystyle{ g(f(x,y),f(x',y')) = f(g(x,x'),g(y,y')) }$ for all $\displaystyle{ x,x',y,y' }$ in $\displaystyle{ X }$. A medial magma is the special case where these operations are equal.