Definition:Inverse Completion

Definition
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 $\circ'$
 * $(3): \quad \gen {S \cup C^{-1} } = \struct {T, \circ'}$

where:
 * $\gen {S \cup C^{-1} }$ 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

 * Construction of Inverse Completion
 * Cancellable Elements of Semigroup form Subsemigroup