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| Format: | Preprint |
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2017
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| Online Access: | https://arxiv.org/abs/1709.00031 |
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| _version_ | 1866909270752100352 |
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| author | Otto, Martin |
| author_facet | Otto, Martin |
| contents | We present a generic construction of finite realisations of amalgamation patterns. An amalgamation pattern is specified by a finite collection of finite template structures together with a collection of partial isomorphisms between them. A realisation is a globally consistent solution to the locally consistent specification of this amalgamation problem: this is a single structure equipped with an atlas of distinguished substructures associated with the template structures, their overlaps realising precisely the identifications induced by the given partial isomorphisms. Our construction is based on natural reduced products with suitable groupoids. The resulting realisations are generic in the sense that they can be made to preserve all symmetries of the specification. They can also be made to be universal w.r.t.\ to local homomorphisms up to any specified size. As key applications of the main construction we discuss finite hypergraph coverings of specified levels of acyclicity and a new route to the lifting of local symmetries to global automorphisms in finite structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1709_00031 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Amalgamation and Symmetry: From Local to Global Consistency in The Finite Otto, Martin Combinatorics 20L05, 05C65, 05E18, 20B25, 20F05, 57M12 We present a generic construction of finite realisations of amalgamation patterns. An amalgamation pattern is specified by a finite collection of finite template structures together with a collection of partial isomorphisms between them. A realisation is a globally consistent solution to the locally consistent specification of this amalgamation problem: this is a single structure equipped with an atlas of distinguished substructures associated with the template structures, their overlaps realising precisely the identifications induced by the given partial isomorphisms. Our construction is based on natural reduced products with suitable groupoids. The resulting realisations are generic in the sense that they can be made to preserve all symmetries of the specification. They can also be made to be universal w.r.t.\ to local homomorphisms up to any specified size. As key applications of the main construction we discuss finite hypergraph coverings of specified levels of acyclicity and a new route to the lifting of local symmetries to global automorphisms in finite structures. |
| title | Amalgamation and Symmetry: From Local to Global Consistency in The Finite |
| topic | Combinatorics 20L05, 05C65, 05E18, 20B25, 20F05, 57M12 |
| url | https://arxiv.org/abs/1709.00031 |