Stable Sub-semimodule of a Series

Given a series defined as $S ⟨ ⟨ Σ^{*} ⟩ ⟩$ this can be thought of as a module we quotient a series $S \in S ⟨ ⟨ Σ^{*} ⟩ ⟩$ with a polynomial $p$ we can write $S = p \cdot S^{'}$ then quotienting $S$ by $p$ gives $S^{'}$ . Any series that can’t be broken down then it dies. So we define the quotient $B$ of $A \subseteq S ⟨ ⟨ Σ^{*} ⟩ ⟩$ with $p$ as:

p^{- 1} A = B = {S^{'} ∣ p \cdot S^{'} \in A}

Definition

We say that a subset $A$ is stable if $p^{- 1} A \subseteq A$ .

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Stable Sub-semimodule of a Series

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