Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914762711891968 |
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| author | Chajda, Ivan Kolařík, Miroslav Länger, Helmut |
| author_facet | Chajda, Ivan Kolařík, Miroslav Länger, Helmut |
| contents | On an arbitrary meet-semilattice S with 0 we define an orthogonality relation and investigate the lattice Cl(S) of all subsets of S closed under this orthogonality. We show that if S is atomic then Cl(S) is a complete atomic Boolean algebra. If S is a pseudocomplemented lattice, this orthogonality relation can be defined by means of the pseudocomplementation. Finally, we show that if S is a complete pseudocomplemented lattice then Cl(S) is a complete Boolean algebra. For pseudocomplemented posets a similar result holds if the subset of pseudocomplements forms a complete lattice satisfying a certain compatibility condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13361 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets Chajda, Ivan Kolařík, Miroslav Länger, Helmut Combinatorics 06B05, 06C15, 06D15, 06D25, 08A11 On an arbitrary meet-semilattice S with 0 we define an orthogonality relation and investigate the lattice Cl(S) of all subsets of S closed under this orthogonality. We show that if S is atomic then Cl(S) is a complete atomic Boolean algebra. If S is a pseudocomplemented lattice, this orthogonality relation can be defined by means of the pseudocomplementation. Finally, we show that if S is a complete pseudocomplemented lattice then Cl(S) is a complete Boolean algebra. For pseudocomplemented posets a similar result holds if the subset of pseudocomplements forms a complete lattice satisfying a certain compatibility condition. |
| title | Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets |
| topic | Combinatorics 06B05, 06C15, 06D15, 06D25, 08A11 |
| url | https://arxiv.org/abs/2404.13361 |