Equivalence of Definitions of Strictly Inductive Semigroup

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Theorem

The following definitions of the concept of Strictly Inductive Semigroup are equivalent:

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.


Proof

$(1)$ implies $(2)$

Let $\struct {S, \circ}$ be a strictly inductive semigroup by definition $1$.

Then by 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.



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Thus $\struct {S, \circ}$ is a strictly inductive semigroup by definition $2$.

$\Box$


$(2)$ implies $(1)$

Let $\struct {S, \circ}$ be a strictly inductive semigroup by definition $2$.

Then by definition $2$:

Let there exist an epimorphism from $\struct {\N_{>0}, +}$ to $\struct {S, \circ}$.


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



This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Thus $\struct {S, \circ}$ is a strictly inductive semigroup by definition $1$.

$\Box$


This needs considerable tedious hard slog to complete it.
In particular: 2 iff 3
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Sources

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