Category:Strictly Inductive Semigroups
This category contains results about Strictly Inductive Semigroups.
Definitions specific to this category can be found in Definitions/Strictly Inductive Semigroups.
Let $\struct {S, \circ}$ be a semigroup.
Definition 1
Let there exist $\beta \in S$ such that the only subset of $S$ containing both $\beta$ and $x \circ \beta$ whenever it contains $x$ is $S$ itself.
- $\exists \beta \in S: \forall A \subseteq S: \paren {\beta \in S \land \paren {\forall x \in A: x \circ \beta \in A} } \implies A = S$
Then $\struct {S, \circ}$ is a strictly inductive semigroup.
Definition 2
Let there exist an epimorphism from $\struct {\N_{>0}, +}$ to $\struct {S, \circ}$.
Then $\struct {S, \circ}$ is a strictly inductive semigroup.
Definition 3
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.
Pages in category "Strictly Inductive Semigroups"
The following 3 pages are in this category, out of 3 total.