Koulman, Jannis (2025) Flats over T-matroids. Bachelor's Thesis, Mathematics.
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Abstract
In this thesis we develop a theory of flats over matroids with coefficients in a tract, and use them to give a novel cryptomorphism of T-matroids. We define T-representations for flats of the underlying matroid. We then use this to show that modular triples of hyperplane functions are linearly dependent if and only if there exists a T-matroid which is a quotient of each of the hyperplane functions. Defining T-flats as the vectors of the T-representation of the flats, we show that for a given T-matroid, the T-flats form a geometric lattice with respect to inclusion, which in fact has the same lattice structure as the lattice of flats of the underlying matroid. Using our previous results, we can give a cryptomorphic definition of a T-matroid in terms of a lattice of T-flats. We conclude the thesis by showing that over a field K, the notion of K-flats coincide with the notion of hyperplane arrangements.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Lorscheid, O. and Top, J. |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 04 Jul 2025 13:00 |
| Last Modified: | 25 Jul 2025 13:48 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/35603 |
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