Definition:Inductive Semigroup

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

Let there exist $\alpha, \beta \in S$ such that the only subset of $S$ containing both $\alpha$ and $x \circ \beta$ whenever it contains $x$ is $S$ itself.

That is:
 * $\exists \alpha, \beta \in S: \forall A \subseteq S: \paren {\alpha \in A \land \paren {\forall x \in A: x \circ \beta \in A} } \implies A = S$

Then $\struct {S, \circ}$ is an inductive semigroup.

Also see

 * Definition:Strictly Inductive Semigroup