# Definition:Inverse Completion

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## 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

- Results about
**inverse completions**can be found**here**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $\S 20$: The Integers