### Abstract

We associate with every commutative continuous semiring S and alphabet S a category whose objects are all sets and a morphism X → Y is determined by a function from X into the semiring of formal series St(Y ∪+ ∗).y of finite words over Y ∪+, an X × Y -matrix over St(Y ∪+∗).y, and a function from X into the continuous St(Y ∪+∗).y-semimodule St(Y ∪+)ωy of series of ω- words over Y ∪+. When S is also an ω-semiring (equipped with an infinite product operation), then we define a fixed point operation over our category and show that it satisfies all identities of iteration categories. We then use this fixed point operation to give semantics to recursion schemes defining series of finite and infinite words. In the particular case when the semiring is the Boolean semiring, we obtain the context-free languages of finite and !-words.

Original language | English |
---|---|

Pages (from-to) | 61-79 |

Number of pages | 19 |

Journal | Acta Cybernetica |

Volume | 23 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2017 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Acta Cybernetica*,

*23*(1), 61-79. https://doi.org/10.14232/actacyb.23.1.2017.5

**Continuous semiring-semimodule pairs and mixed algebraic systems.** / Ésik, Z.; Kuich, Werner.

Research output: Contribution to journal › Article

*Acta Cybernetica*, vol. 23, no. 1, pp. 61-79. https://doi.org/10.14232/actacyb.23.1.2017.5

}

TY - JOUR

T1 - Continuous semiring-semimodule pairs and mixed algebraic systems

AU - Ésik, Z.

AU - Kuich, Werner

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We associate with every commutative continuous semiring S and alphabet S a category whose objects are all sets and a morphism X → Y is determined by a function from X into the semiring of formal series St(Y ∪+ ∗).y of finite words over Y ∪+, an X × Y -matrix over St(Y ∪+∗).y, and a function from X into the continuous St(Y ∪+∗).y-semimodule St(Y ∪+)ωy of series of ω- words over Y ∪+. When S is also an ω-semiring (equipped with an infinite product operation), then we define a fixed point operation over our category and show that it satisfies all identities of iteration categories. We then use this fixed point operation to give semantics to recursion schemes defining series of finite and infinite words. In the particular case when the semiring is the Boolean semiring, we obtain the context-free languages of finite and !-words.

AB - We associate with every commutative continuous semiring S and alphabet S a category whose objects are all sets and a morphism X → Y is determined by a function from X into the semiring of formal series St(Y ∪+ ∗).y of finite words over Y ∪+, an X × Y -matrix over St(Y ∪+∗).y, and a function from X into the continuous St(Y ∪+∗).y-semimodule St(Y ∪+)ωy of series of ω- words over Y ∪+. When S is also an ω-semiring (equipped with an infinite product operation), then we define a fixed point operation over our category and show that it satisfies all identities of iteration categories. We then use this fixed point operation to give semantics to recursion schemes defining series of finite and infinite words. In the particular case when the semiring is the Boolean semiring, we obtain the context-free languages of finite and !-words.

UR - http://www.scopus.com/inward/record.url?scp=85020074641&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020074641&partnerID=8YFLogxK

U2 - 10.14232/actacyb.23.1.2017.5

DO - 10.14232/actacyb.23.1.2017.5

M3 - Article

AN - SCOPUS:85020074641

VL - 23

SP - 61

EP - 79

JO - Acta Cybernetica

JF - Acta Cybernetica

SN - 0324-721X

IS - 1

ER -