### Abstract

We have seen that, with respect to homomorphic realization, the ν_{i}—products behave in a way similar to the α_{i}—products on classes satisfying the Letičevskiî criterion or not satisfying the Letičevskiî criteria. In particular, a class K is homomorphically ν_{3}—complete if and only if it satisfies the Letičevskiî criterion. As opposed to the α_{i}—products, the ν_{i}—hierarchy is proper on classes with the semi-Letičevskiî criterion. This holds also for homomorphic realization. To give a summary of the rest of our comparison results, take any two of our product notions, the “β—product” and the “γ—product”, say. Define γ if holds for all K. Similarly let γ if we have for all K. We obtain two poset structures whose exact diagrams are given in the figures below. The bottom is the quasi-direct product in both cases, for it is obvious that and ν_{1}, henceforth also and. (We write β<γ if β≤γ but γ≰β.) Recently it has been shown by the first two authors that there is a class K satisfying the Letičevskiî criterion but which is not homomorphically ν_{2}-complete.

Original language | English |
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Title of host publication | Fundamentals of Computation Theory - International Conference, FCT 1989, Proceedings |

Publisher | Springer Verlag |

Pages | 137-144 |

Number of pages | 8 |

ISBN (Print) | 9783540514985 |

DOIs | |

Publication status | Published - Jan 1 1989 |

Event | 7th International Conference on Fundamentals of Computation Theory, FCT 1989 - Szeged, Hungary Duration: Aug 21 1989 → Aug 25 1989 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 380 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 7th International Conference on Fundamentals of Computation Theory, FCT 1989 |
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Country | Hungary |

City | Szeged |

Period | 8/21/89 → 8/25/89 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Fundamentals of Computation Theory - International Conference, FCT 1989, Proceedings*(pp. 137-144). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 380 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-51498-8_13

**On product hierarchies of automata.** / Dömösi, P.; Ésik, Z.; Imreh, B.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Fundamentals of Computation Theory - International Conference, FCT 1989, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 380 LNCS, Springer Verlag, pp. 137-144, 7th International Conference on Fundamentals of Computation Theory, FCT 1989, Szeged, Hungary, 8/21/89. https://doi.org/10.1007/3-540-51498-8_13

}

TY - GEN

T1 - On product hierarchies of automata

AU - Dömösi, P.

AU - Ésik, Z.

AU - Imreh, B.

PY - 1989/1/1

Y1 - 1989/1/1

N2 - We have seen that, with respect to homomorphic realization, the νi—products behave in a way similar to the αi—products on classes satisfying the Letičevskiî criterion or not satisfying the Letičevskiî criteria. In particular, a class K is homomorphically ν3—complete if and only if it satisfies the Letičevskiî criterion. As opposed to the αi—products, the νi—hierarchy is proper on classes with the semi-Letičevskiî criterion. This holds also for homomorphic realization. To give a summary of the rest of our comparison results, take any two of our product notions, the “β—product” and the “γ—product”, say. Define γ if holds for all K. Similarly let γ if we have for all K. We obtain two poset structures whose exact diagrams are given in the figures below. The bottom is the quasi-direct product in both cases, for it is obvious that and ν1, henceforth also and. (We write β<γ if β≤γ but γ≰β.) Recently it has been shown by the first two authors that there is a class K satisfying the Letičevskiî criterion but which is not homomorphically ν2-complete.

AB - We have seen that, with respect to homomorphic realization, the νi—products behave in a way similar to the αi—products on classes satisfying the Letičevskiî criterion or not satisfying the Letičevskiî criteria. In particular, a class K is homomorphically ν3—complete if and only if it satisfies the Letičevskiî criterion. As opposed to the αi—products, the νi—hierarchy is proper on classes with the semi-Letičevskiî criterion. This holds also for homomorphic realization. To give a summary of the rest of our comparison results, take any two of our product notions, the “β—product” and the “γ—product”, say. Define γ if holds for all K. Similarly let γ if we have for all K. We obtain two poset structures whose exact diagrams are given in the figures below. The bottom is the quasi-direct product in both cases, for it is obvious that and ν1, henceforth also and. (We write β<γ if β≤γ but γ≰β.) Recently it has been shown by the first two authors that there is a class K satisfying the Letičevskiî criterion but which is not homomorphically ν2-complete.

UR - http://www.scopus.com/inward/record.url?scp=30844470452&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30844470452&partnerID=8YFLogxK

U2 - 10.1007/3-540-51498-8_13

DO - 10.1007/3-540-51498-8_13

M3 - Conference contribution

SN - 9783540514985

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 137

EP - 144

BT - Fundamentals of Computation Theory - International Conference, FCT 1989, Proceedings

PB - Springer Verlag

ER -