Conditions under which Commutative Semigroup is Group/Statement of Conditions

Statement of Conditions under which Commutative Semigroup is Group

Let $\struct {S, \circ}$ be a commutative semigroup.

Let $\struct {S, \circ}$ have the following properties:

$(1)$	$:$			$\ds \forall x \in S: \exists y \in S:$	$\ds y \circ x = x $
$(2)$	$:$			$\ds \forall x, y \in S:$	$\ds y \circ x = x \implies \exists z \in S: z \circ x = y $

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.16$

\((1)\)	$:$			\(\ds \forall x \in S: \exists y \in S:\)	\(\ds y \circ x = x \)
\((2)\)	$:$			\(\ds \forall x, y \in S:\)	\(\ds y \circ x = x \implies \exists z \in S: z \circ x = y \)