Automorphic inverse property loop

Definition
A loop is said to satisfy the automorphic inverse property if the following two conditions are satisfied:


 * It is a defining ingredient::inverse property loop: every element in the algebra loop has a well-defined inverse that plays the role of both a left and right inverse. In other words, for every $$x$$, there is an element $$x^{-1}$$ such that $$x^{-1} * (x * y) = (y * x) * x^{-1} = y$$ for all $$y$$ in the loop.
 * if $$x^{-1}$$ denotes the inverse of $$x$$, then:

$$\! (x * y)^{-1} = x^{-1} * y^{-1}$$