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## Congruence Lattices of Finite Algebras

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PhD_2012_DeMeo_uh.pdf | Version for UH users | 974.21 kB | Adobe PDF | View/Open |

PhD_2012_DeMeo_r.pdf | Version for non-UH users. Copying/Printing is not permitted | 976.28 kB | Adobe PDF | View/Open |

### Item Summary

Title: | Congruence Lattices of Finite Algebras |

Authors: | DeMeo, William J. |

Advisor: | Freese, Ralph |

Issue Date: | 2012 |

Publisher: | University of Hawaii at Manoa |

Abstract: | An important and long-standing open problem in universal algebra asks whether every finite lattice is isomorphic to the congruence lattice of a finite algebra. Until this problem is resolved, our understanding of finite algebras is incomplete, since, given an arbitrary finite algebra, we cannot say whether there are any restrictions on the shape of its congruence lattice. If we find a finite lattice that does not occur as the congruence lattice of a finite algebra (as many suspect we will), then we can finally declare that such restrictions do exist. By a well known result of Palfy and Pudlak, the problem would be solved if we could prove the existence of a finite lattice that is not the congruence lattice of a transitive group action or, equivalently, is not an interval in the lattice of subgroups of a finite group. Thus the problem of characterizing congruence lattices of finite algebras is closely related to the problem of characterizing intervals in subgroup lattices. In this work, we review a number of methods for finding a finite algebra with a given congruence lattice, including searching for intervals in subgroup lattices. We also consider methods for proving that algebras with a given congruence lattice exist without actually constructing them. By combining these well known methods with a new method we have developed, and with much help from computer software like the UACalc and GAP, we prove that with one possible exception every lattice with at most seven elements is isomorphic to the congruence lattice of a finite algebra. As such, we have identified the unique smallest lattice for which there is no known representation. We examine this exceptional lattice in detail, and prove results that characterize the class of algebras that could possibly represent this lattice. We conclude with what we feel are the most interesting open questions surrounding this problem and discuss possibilities for future work. |

Description: | Thesis (Ph. D.)--University of Hawaii at Manoa, 2012. |

Pages/Duration: | viii, 122 leaves |

URI/DOI: | http://hdl.handle.net/10125/25938 |

Rights: | All UHM dissertations and theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission from the copyright owner. |

Appears in Collections: | Ph.D. - Mathematics Ph.D. Dissertations- Mathematics Department |

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