The fundamental theorem of finite semidistributive lattices
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866911674488848384 |
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| author | Reading, Nathan Speyer, David E Thomas, Hugh |
| author_facet | Reading, Nathan Speyer, David E Thomas, Hugh |
| contents | We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only if there exists a set Sha with some additional structure, such that L is isomorphic to the admissible subsets of Sha ordered by inclusion; in this case, Sha and its additional structure are uniquely determined by L." The additional structure on Sha is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_08050 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The fundamental theorem of finite semidistributive lattices Reading, Nathan Speyer, David E Thomas, Hugh Combinatorics Representation Theory 06B05, 06A15, 06B15, 06D75 We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only if there exists a set Sha with some additional structure, such that L is isomorphic to the admissible subsets of Sha ordered by inclusion; in this case, Sha and its additional structure are uniquely determined by L." The additional structure on Sha is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices. |
| title | The fundamental theorem of finite semidistributive lattices |
| topic | Combinatorics Representation Theory 06B05, 06A15, 06B15, 06D75 |
| url | https://arxiv.org/abs/1907.08050 |