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

• Results about strictly inductive semigroups can be found here.

Sources

• 1960: Seth Warner: Mathematical Induction in Commutative Semigroups (Amer. Math. Monthly Vol. 67, no. 6: pp. 533 – 537)  www.jstor.org/stable/2309170
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.8$