On certain closure operators defined by families of semiring morphisms (Q1305060)

Let \((A,+,\cdot)\) be a continuous semiring and define the star \(a^*\) for every \(a\in A\) by \(a^*=\sum_{i\geq 0} a^i\) as usual. A subset \(\overline A\) of \(A\) is fully rationally closed if \(a^*\in\overline A\) for every \(a\in\overline A\), and \({\mathfrak{Rat}}(B)\) denotes the smallest fully rationally closed subsemiring of \(A\) containing \(B\subseteq A\). Moreover, let \(Q_\infty\) be some fixed countably infinite index set satisfying \(q_1,q_2\Rightarrow(q_1,q_2)\in Q_\infty\), and \(\mathfrak H\) a nonempty subfamily of the family of all morphisms \(h\colon A\to A^{Q\times Q}\), \(Q\subset Q_\infty\), \(Q\) finite. A subsemiring \(B\) of \(A\) is called \(\mathfrak H\)-closed if \(\{h(a)_{q_1,q_2}\mid a\in B,\;h\colon A\to A^{Q\times Q}\) in \({\mathfrak H},\;q_1,q_2\in Q\}\subseteq B\) and \({\mathfrak{Rat}}(B)=B\). Under certain conditions, such \(\mathfrak H\)-closed subsemirings \(B\) can be characterized by \({\mathfrak{Rat}}(A')\)-algebraic systems for some subset \(A'\subseteq A\). These results can be applied to formal power series to obtain strong ``normal forms'' for abstract families of power series and languages.