Free iterative and iteration \(K\)-semialgebras
type: Item
instance of: scholarly article
title: Free iterative and iteration \(K\)-semialgebras
zbMATH Open document ID: 1260.08001
published in: Algebra Universalis
publication date: 2012-05-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.1507
review text: 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.
reviewed by: Saeed Salehi
Mathematics Subject Classification ID: 08A70
Mathematics Subject Classification ID: 68Q70
Mathematics Subject Classification ID: 16Y60
zbMATH DE Number: 6029773
zbMATH Keywords: rational power series
zbMATH Keywords: partial iteration semiring
zbMATH Keywords: iterative semiring
zbMATH Keywords: partial iteration \(K\)-semialgebra
zbMATH Keywords: iterative \(K\)-semialgebra
author: Zoltán Ésik
author: W. Kuich
MaRDI profile type: Publication
OpenAlex ID: W1616653757
arXiv ID: 1008.1507
cites work: Conjugacy and Equivalence of Weighted Automata and Functional Transducers
cites work: Q3056218
cites work: Q5287508
cites work: Axiomatizing rational power series over natural numbers
cites work: Q5506336
cites work: Q3934450
cites work: Q5639639
cites work: On Kleene algebras
cites work: Q4079524
cites work: Simple program schemes and formal languages
cites work: Group axioms for iteration
cites work: Inductive \(^{*}\)-semirings
cites work: Cycle-Free Finite Automata in Partial Iterative Semirings
cites work: Q4433737
cites work: Q4229053
cites work: A completeness theorem for Kleene algebras and the algebra of regular events
cites work: Q3975995
cites work: Complete systems of \(\mathcal B\)-rational identities
cites work: THE EQUALITY PROBLEM FOR RATIONAL SERIES WITH MULTIPLICITIES IN THE TROPICAL SEMIRING IS UNDECIDABLE
cites work: A complete axiom system for rational sets with multiplicity
cites work: Q3644388
cites work: Two Complete Axiom Systems for the Algebra of Regular Events
cites work: On the definition of a family of automata
cites work: On a Theorem of R. Jungen
DOI: 10.1007/S00012-012-0179-Y
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description: scientific article; zbMATH DE number 6029773