Elements of algebraic geometry and the positive theory of partially commutative groups
The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories...
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| Format: | Journal article |
| Language: | English |
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2010
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| author | Casals-Ruiz, M Kazachkov, I |
| author_facet | Casals-Ruiz, M Kazachkov, I |
| author_sort | Casals-Ruiz, M |
| collection | OXFORD |
| description | The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories of the direct factors of G (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group H. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of H has quantifier elimination and that arbitrary first-order formulas lift from H to H * F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable. ©Canadian Mathematical Society 2010. |
| first_indexed | 2024-03-06T21:30:25Z |
| format | Journal article |
| id | oxford-uuid:4483f525-27c7-4d9e-9038-4c9f0cd35b60 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:30:25Z |
| publishDate | 2010 |
| record_format | dspace |
| spelling | oxford-uuid:4483f525-27c7-4d9e-9038-4c9f0cd35b602022-03-26T15:01:58ZElements of algebraic geometry and the positive theory of partially commutative groupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4483f525-27c7-4d9e-9038-4c9f0cd35b60EnglishSymplectic Elements at Oxford2010Casals-Ruiz, MKazachkov, IThe first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories of the direct factors of G (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group H. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of H has quantifier elimination and that arbitrary first-order formulas lift from H to H * F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable. ©Canadian Mathematical Society 2010. |
| spellingShingle | Casals-Ruiz, M Kazachkov, I Elements of algebraic geometry and the positive theory of partially commutative groups |
| title | Elements of algebraic geometry and the positive theory of partially commutative groups |
| title_full | Elements of algebraic geometry and the positive theory of partially commutative groups |
| title_fullStr | Elements of algebraic geometry and the positive theory of partially commutative groups |
| title_full_unstemmed | Elements of algebraic geometry and the positive theory of partially commutative groups |
| title_short | Elements of algebraic geometry and the positive theory of partially commutative groups |
| title_sort | elements of algebraic geometry and the positive theory of partially commutative groups |
| work_keys_str_mv | AT casalsruizm elementsofalgebraicgeometryandthepositivetheoryofpartiallycommutativegroups AT kazachkovi elementsofalgebraicgeometryandthepositivetheoryofpartiallycommutativegroups |