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.

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