Definition:Strictly Inductive Semigroup/Definition 3
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Definition
Let $\struct {S, \circ}$ be a semigroup.
Let $\struct {S, \circ}$ be such that either:
- $\struct {S, \circ}$ is isomorphic to $\struct {\N_{>0}, +}$
or:
- there exist $m, n \in \N_{>0}$ such that $\struct {S, \circ}$ is isomorphic to $\struct {\map {D^*} {m, n}, +^*_{m, n} }$
where $\struct {\map {D^*} {m, n}, +^*_{m, n} }$ is the restricted dipper semigroup on $\tuple {m, n}$.
Then $\struct {S, \circ}$ is a strictly inductive semigroup.
Also see
- Results about strictly inductive semigroups can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.8 \ 3^\circ$