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.