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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2006
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/0609114 |
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| _version_ | 1866917354291593216 |
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| author | Douka, P. Felouzis, V. |
| author_facet | Douka, P. Felouzis, V. |
| contents | We investigate the representation of lattices as sublattices of the lattice of all convex subsets (intervals) of a linearly ordered set $(X,\le)$. We introduce the purely lattice-theoretic notion of a \textit{loc-lattice} and prove that every loc-lattice is representable as a lattice of intervals. Furthermore, we provide the complete, unabridged construction for the general representation theorem, establishing that a well-separated lattice is faithfully representable as a lattice of intervals if and only if it is a loc-lattice. Finally, we apply these results to general topology, obtaining novel algebraic characterizations for the bases of weakly orderable and completely orderable topological spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0609114 |
| institution | arXiv |
| publishDate | 2006 |
| record_format | arxiv |
| spellingShingle | Lattices which can be represented as lattices of intervals Douka, P. Felouzis, V. General Mathematics Representation Theory Primary 06B15, Secondary 06B30, 54F05 We investigate the representation of lattices as sublattices of the lattice of all convex subsets (intervals) of a linearly ordered set $(X,\le)$. We introduce the purely lattice-theoretic notion of a \textit{loc-lattice} and prove that every loc-lattice is representable as a lattice of intervals. Furthermore, we provide the complete, unabridged construction for the general representation theorem, establishing that a well-separated lattice is faithfully representable as a lattice of intervals if and only if it is a loc-lattice. Finally, we apply these results to general topology, obtaining novel algebraic characterizations for the bases of weakly orderable and completely orderable topological spaces. |
| title | Lattices which can be represented as lattices of intervals |
| topic | General Mathematics Representation Theory Primary 06B15, Secondary 06B30, 54F05 |
| url | https://arxiv.org/abs/math/0609114 |