{"title":"可分刚性群中的通用类型和通用元素","authors":"A. G. Myasnikov, N. S. Romanovskii","doi":"10.1007/s10469-023-09726-x","DOIUrl":null,"url":null,"abstract":"<p>A group <i>G</i> is said to be m-rigid if it contains a normal series of the form <i>G</i> = <i>G</i><sub>1</sub> > <i>G</i><sub>2</sub> > . . . > <i>G</i><sub><i>m</i></sub> > <i>G</i><sub><i>m+</i>1</sub> = 1, whose quotients <i>G</i><sub><i>i</i></sub>/<i>G</i><sub><i>i+</i>1</sub> are Abelian and, treated as (right) ℤ[<i>G</i>/<i>G</i><sub><i>i</i></sub>]-modules, are torsion-free. A rigid group <i>G</i> is said to be divisible if elements of the quotient <i>ρ</i><sub><i>i</i></sub>(<i>G</i>)/<i>ρ</i><sub><i>i+</i>1</sub>(<i>G</i>) are divisible by nonzero elements of the ring ℤ[<i>G</i>/<i>ρ</i><sub><i>i</i></sub>(<i>G</i>)]. Previously, it was proved that the theory of divisible m-rigid groups is complete and ω-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible m-rigid group <i>G</i>.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generic Types and Generic Elements in Divisible Rigid Groups\",\"authors\":\"A. G. Myasnikov, N. S. Romanovskii\",\"doi\":\"10.1007/s10469-023-09726-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A group <i>G</i> is said to be m-rigid if it contains a normal series of the form <i>G</i> = <i>G</i><sub>1</sub> > <i>G</i><sub>2</sub> > . . . > <i>G</i><sub><i>m</i></sub> > <i>G</i><sub><i>m+</i>1</sub> = 1, whose quotients <i>G</i><sub><i>i</i></sub>/<i>G</i><sub><i>i+</i>1</sub> are Abelian and, treated as (right) ℤ[<i>G</i>/<i>G</i><sub><i>i</i></sub>]-modules, are torsion-free. A rigid group <i>G</i> is said to be divisible if elements of the quotient <i>ρ</i><sub><i>i</i></sub>(<i>G</i>)/<i>ρ</i><sub><i>i+</i>1</sub>(<i>G</i>) are divisible by nonzero elements of the ring ℤ[<i>G</i>/<i>ρ</i><sub><i>i</i></sub>(<i>G</i>)]. Previously, it was proved that the theory of divisible m-rigid groups is complete and ω-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible m-rigid group <i>G</i>.</p>\",\"PeriodicalId\":7422,\"journal\":{\"name\":\"Algebra and Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-01-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebra and Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10469-023-09726-x\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra and Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10469-023-09726-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
摘要
如果一个群 G 包含一个形式为 G = G1 > G2 > ... > Gm > Gm+1 = 1 的正序列，其商数 Gi/Gi+1 是阿贝尔的，并且作为（右）ℤ[G/Gi]模块处理时是无扭的，那么这个群 G 可以说是 m 刚群。如果商ρi(G)/ρi+1(G)中的元素能被ℤ[G/ρi(G)]环中的非零元素整除，则称刚性群 G 是可分的。在此之前，我们已经证明了可分 m-rigid 群理论是完整且 ω 稳定的。在本文中，我们给出了可分 m-rigid 群 G 上通用元素和类型的代数描述。
Generic Types and Generic Elements in Divisible Rigid Groups
A group G is said to be m-rigid if it contains a normal series of the form G = G1 > G2 > . . . > Gm > Gm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, treated as (right) ℤ[G/Gi]-modules, are torsion-free. A rigid group G is said to be divisible if elements of the quotient ρi(G)/ρi+1(G) are divisible by nonzero elements of the ring ℤ[G/ρi(G)]. Previously, it was proved that the theory of divisible m-rigid groups is complete and ω-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible m-rigid group G.
期刊介绍：
This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions.
Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences.
All articles are peer-reviewed.
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