On countable isotypic structures. (English) Zbl 07910109
Summary: We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with countable underlying sets: totally ordered sets, fields, and groups. This answers an old question by B. Plotkin for groups.
References:
| [1] | E. Bunina. Isomorphisms and elementary equivalence of Chevalley groups over com-mutative rings. Sb. Math., 210(8):1067-1091, 2019. · Zbl 1472.20110 |
| [2] | E. Bunina. Isotypic equivalence of abelian 𝑝-groups with separable reduced parts. 2024. Preprint: https://arxiv.org/abs/2402.11261. |
| [3] | E. Yu. Daniyarova, A. G. Myasnikov, and V. N. Remeslennikov. Algebraic geometry over algebraic structures. II. Foundations. J. Math. Sci, 185:389-416, 2012. · Zbl 1256.03037 |
| [4] | A. J. Engler and A. Prestel. Valued Fields. Springer, Berlin et al., 2005. · Zbl 1128.12009 |
| [5] | M. D. Fried and M. Jarden. Field Arithmetic, 3nd ed. Springer Verlag, Berlin et al., 2008. · Zbl 1145.12001 |
| [6] | D. Marker. Model Theory: An Introduction. Springer, New York, 2002. · Zbl 1003.03034 |
| [7] | A. G. Myasnikov and V. Remeslennikov. Algebraic geometry over groups II, Logical foundations. J. Algebra, 234:225-276, 2000. · Zbl 0970.20017 |
| [8] | A. G. Myasnikov and N. S. Romanovskii. Characterization of finitely generated groups by types. Internat. J. Algebra Comput., 28(8):1613-1632, 2018. · Zbl 1483.20055 |
| [9] | J. Pas. Uniform 𝑝-adic cell decomposition and local zeta functions. J. Reine Angew. Math., 399:137-172, 1989. · Zbl 0666.12014 |
| [10] | B. Plotkin. Isotyped algebras. Proceedings of the Steklov Institute of Mathematics, 278:91-115, 2012. · Zbl 1361.03067 |
| [11] | B. Plotkin. Algebraic logic and logical geometry in arbitrary varieties of algebras. In Proceedings of the Conf. on Group Theory, Combinatorics and Computing, AMS Contemporary Math. series, pages 151-169, 2014. · Zbl 1301.08011 |
| [12] | B. Plotkin. Seven lectures on universal algebraic geometry. Groups, Algebras and Identities, AMS, Contemporary Mathematics, 726:143-217, 2019. Israel Mathematical Conferences Proceedings. |
| [13] | B. Plotkin, E. Aladova, and E. Plotkin. Algebraic logic and logically-geometric types in varieties of algebras. Journal of Algebra and its Applications, 12, 2012. Paper No. 1250146. · Zbl 1297.03022 |
| [14] | B. Plotkin and E. Plotkin. Multi-sorted logic and logical geometry: some problems. Demonstratio Math., 48:577-618, 2015. |
| [15] | B. Plotkin and G. Zhitomirski. Some logical invariants of algebras and logical relations between algebras. St. Peterburg Math. J., 19:859-879, 2008. |
| [16] | B. Poizat. A Course in Model Theory: An Introduction to Contemporary Mathematical Logic. Springer, Berlin et al., 2000. · Zbl 0951.03002 |
| [17] | G. Zhitomirski. On types of points and algebras. International Journal of Algebra and Computation, 28(8):1717-1730, 2018. · Zbl 1509.03093 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
