Definition:Inverse Completion

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

Let $\struct {C, \circ} \subseteq \struct {S, \circ}$ be the subsemigroup of cancellable elements of $\struct {S, \circ}$.

Let $\struct {T, \circ'}$ be a semigroup defined such that:

$(1): \quad \struct {S, \circ}$ is a subsemigroup of $\struct {T, \circ'}$
$(2): \quad$ Every element of $C$ has an inverse in $T$ for $\circ'$
$(3): \quad \gen {S \cup C^{-1} } = \struct {T, \circ'}$


$\gen {S \cup C^{-1} }$ denotes the subsemigroup of $\struct {T, \circ'}$ generated by $S \cup C^{-1}$
$C^{-1}$ denotes the inverse of $C$.

Then $\struct {T, \circ'}$ is called an inverse completion of $\struct {S, \circ}$.

Also known as

Some sources use a hyphen: inverse-completion.

Some sources call it an involution-monoid.

Also see

  • Results about inverse completions can be found here.