Auto magma object

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