Free iterative and iteration \(K\)-semialgebras
Let \(K^{\text{rat}}\langle\!\langle\Sigma^\ast\rangle\!\rangle\) denote the set of rational power series over the finite alphabet \(\Sigma\) with coefficients in the semiring \(K\). Equipped with a partial star operation, this set is both a partial iteration semiring and an iterative semiring. When \(K\) is commutative, \(K^{\text{rat}}\langle\!\langle\Sigma^\ast\rangle\!\rangle\) is a partial iteration \(K\)-semialgebra and also an iterative \(K\)-semialgebra, and when \(K\) is also iterative (i.e., when \(K\) is a commutative iteration semiring) then \(K^{\text{rat}}\langle\!\langle\Sigma^\ast\rangle\!\rangle\) is an iteration \(K\)-semialgebra. In this paper the semiring \(K^{\text{rat}}\langle\!\langle\Sigma^\ast\rangle\!\rangle\) is characterized as: {\parindent=6mm \begin{itemize}\item[--] For a proper commutative semiring \(K\), \(K^{\text{rat}}\langle\!\langle\Sigma^\ast\rangle\!\rangle\) is a free iterative \(K\)-semialgebra on \(\Sigma\). (Theorem 7.1) \item[--] For a proper commutative semiring \(K\), \(K^{\text{rat}}\langle\!\langle\Sigma^\ast\rangle\!\rangle\) is a free symmetric iterative \(K\)-semialgebra on \(\Sigma\). (Corollary 7.2) \item[--] For a commutative semiring \(K\) every finitely generated subsemiring of which is contained in a Noetherian subsemiring, \(K^{\text{rat}}\langle\!\langle\Sigma^\ast\rangle\!\rangle\) is a free (symmetric) iterative \(K\)-semialgebra on \(\Sigma\). (Corollary 7.3) \item[--] For a strongly proper atomistic commutative semiring \(K\), \(K^{\text{rat}}\langle\!\langle\Sigma^\ast\rangle\!\rangle\) is a free partial iteration \(K\)-semialgebra on \(\Sigma\). (Theorem 8.5) \item[--] For a proper atomistic commutative semiring \(K\) which is additively generated by its units, \(K^{\text{rat}}\langle\!\langle\Sigma^\ast\rangle\!\rangle\) is a free partial iteration \(K\)-semialgebra on \(\Sigma\). (Corollary 8.6) \end{itemize}} The paper contains also some other results.
scientific article; zbMATH DE number 6029773
scientific article