Conditions under which Commutative Semigroup is Group/Lemma 2

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Lemma for Conditions under which Commutative Semigroup is Group

Suppose the following:

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


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

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


Then:

If $y \circ x = x$, then $y \circ y = y$.


Proof

\(\ds y \circ x\) \(=\) \(\ds x\) by hypothesis
\(\ds \leadsto \ \ \) \(\ds y \circ \paren {y \circ x}\) \(=\) \(\ds y \circ x\) premultiplying both sides by $y$
\(\ds \) \(=\) \(\ds x\) by hypothesis
\(\ds \leadsto \ \ \) \(\ds \paren {y \circ y} \circ x\) \(=\) \(\ds x\) Semigroup Axiom $\text S 1$: Associativity
\(\ds \leadsto \ \ \) \(\ds y \circ y\) \(=\) \(\ds y\) Lemma 1

$\blacksquare$


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