Definition:Inverse Completion

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Definition

Let $\left({S, \circ}\right)$ be a semigroup.

Let $\left ({C, \circ}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$.

Let $\left({T, \circ'}\right)$ be defined such that:

$(1): \quad \left({S, \circ}\right)$ is a subsemigroup of $\left({T, \circ'}\right)$

$(2): \quad$ Every element of $C$ has an inverse in $T$ for $\circ'$

$(3): \quad \left\langle{S \cup C^{-1}}\right\rangle = \left({T, \circ'}\right)$

where $\left\langle{S \cup C^{-1}}\right\rangle$ denotes the semigroup generated by $S \cup C^{-1}$.

Then $\left({T, \circ'}\right)$ is called an inverse completion of $\left({S, \circ}\right)$.