## Abstract

Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of boolean representable simplicial complexes. This turns out to be the direct sum of a uniform matroid with the lift matroid of the complete gain link graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar gain graph. This gives a new perspective on both matroids and also an application of matroid theory to the theory of finite semigroups.

Original language | English |
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Pages (from-to) | 55-91 |

Number of pages | 37 |

Journal | Journal of Algebra |

Volume | 578 |

DOIs | |

State | Published - 15 Jul 2021 |

### Bibliographical note

Publisher Copyright:© 2021 Elsevier Inc.

### Funding

The first author acknowledges support from the Binational Science Foundation (BSF) of the United States and Israel, grant number 2012080. The second author acknowledges support from the Simons Foundation (Simons Travel Grant Number 313548). The third author was partially supported by CMUP, which is financed by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P. under the project with reference UIDB/00144/2020.

Funders | Funder number |
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United States and Israel | 2012080 |

Simons Foundation | 313548 |

United States-Israel Binational Science Foundation | |

Fundação para a Ciência e a Tecnologia | UIDB/00144/2020 |

Centro de Matemática Universidade do Porto |

## Keywords

- Boolean representable simplicial complex
- Brandt groupoid
- Dowling lattice
- Lift matroid
- Matroid
- Rhodes lattice
- Wreath product