# Category:Inverse Completions

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This category contains results about **Inverse Completions**.

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 $\struct {S, \circ}$.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

### C

### E

### F

## Pages in category "Inverse Completions"

The following 22 pages are in this category, out of 22 total.

### E

### I

- Identity of Inverse Completion of Commutative Monoid
- Inverse Completion is Commutative Monoid
- Inverse Completion is Commutative Semigroup
- Inverse Completion is Unique
- Inverse Completion Less Zero of Integral Domain is Closed
- Inverse Completion of Commutative Semigroup is Abelian Group
- Inverse Completion of Commutative Semigroup is Inverse Completion of Itself
- Inverse Completion of Integral Domain Exists
- Inverse Completion of Natural Numbers
- Inverse Completion Theorem
- Inverse Element in Inverse Completion of Commutative Monoid