Definition:Strictly Inductive Semigroup/Definition 1

Definition
Let $\struct {S, \circ}$ be a semigroup. 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.

Also see

 * Equivalence of Definitions of Strictly Inductive Semigroup