Recently i studied the book Fundamental of semigroup theory by John M.Howie and i go through some basic result and i found proposition related to zero semigroup which is
in the proof when we moved to the converse part actually we have to prove that $G^0$ is a semigroup but in the text author prove $G$ is a group. is there any point which i did not understand
The author proves that $G=S\setminus\{0\}$ is a group, and so $S=G\cup\{0\}$ is an $0$-group by the definition at the top. We have $S=G_0$ again by this definition.