Definition:Strictly Inductive Semigroup/Definition 3

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

• Equivalence of Definitions of Strictly Inductive Semigroup

• 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 $16.8 \ 3^\circ$