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