Definition:Inverse Semigroup

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An inverse semigroup is a semigroup $\left({S, \circ}\right)$ such that:

$\forall a \in S: \exists! b \in S: a = a \circ b \circ a, b = b \circ a \circ b$


Let $a\in S$.

The inverse of $a$ is the unique element $b\in S$ such that:

$a = a \circ b \circ a$ and $b = b \circ a \circ b$

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