Is there a name for the generalization of the concept "Abelian group" where the axiom $-x+x = 0$ is weakened to the following? Is there a name for the generalization of the concept "Abelian group" where the axiom $−x+x=0$ is replaced by the following list?


*

*$−0=0$

*$−(x+y)=−x+−y$

*$−(−x)=x$

*$x+(-x)+x = x$


In multiplicative notation; we replace the axiom $x^{-1}x=1$ with the following list:


*

*$1^{-1}=1$

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

*$(x^{-1})^{-1}=x$

*$xx^{-1}x = x$


Examples. 


*

*Any Abelian group satisfies the above axioms in their additive form.

*The multiplicative structure of any zero-totalized field satisfies the above axioms in their multiplicative form, but does not satisfy $x^{-1}x=1$, since $0^{-1} \cdot 0 = 0 \cdot 0 = 0$.


Apparently it's called a (commutative) inverse monoid. For further details, see Wikipedia1, 2, 3 or Lawson's Inverse Semigroups4.
(I haven't proven that the sets of axioms are equivalent. You may want to reserve the bounty for someone who does so.)
