Semigroup is Group Iff Latin Square Property Holds/Proof 1

Necessary Condition
Let $\struct {S, \circ}$ be a group.

$\struct {S, \circ}$ is a semigroup by the definition of a group.

By Group has Latin Square Property, the Latin square property holds in $S$.

Sufficient Condition
Let $\struct {S, \circ}$ be a semigroup on which the Latin square property holds.

Taking the group axioms in turn:

$\struct {S, \circ}$ is semigroup.

Hence $\struct {S, \circ}$ is closed by definition.

$\struct {S, \circ}$ is semigroup.

Hence $\circ$ is associative by definition.

Let $a \in S$.

Then there is an $x \in S$ such that:

Let $b \in S$.

Then there is a $y \in S$ such that:

Proving that $x$ is a right identity:

Let $c \in S$.

Then there is a $z \in S$ such that:

Proving that $x$ is a left identity and thus an identity:

Relabel $x$ as $e$.

Let $x \in S$.

Then $x$ has a right and left inverse by the property as there exists a $y$ and $z$ in $S$ such that:

We shall prove that $y$ is also the left inverse of $x$:

Hence every element of $S$ has a inverse.