论文标题
双线性形式的矢量空间中可定义的集合，组和字段
Sets, groups, and fields definable in vector spaces with a bilinear form
论文作者
论文摘要
我们在理论上研究了无限二维矢量空间的理论中的可定义集，组和田地，该载体是在配备非字体对称（或交替的）双线性形式的代数封闭场上。首先，我们定义一个（$ \ mathbb {n} \ times \ times \ mathbb {z}，\ leq_ {lex} $） - 在$ t_ \ infty $中可定义的集合上的有价值的维度，享受了Morley等级的许多属性，这些属性都以极大的最小理论。然后，使用此维度概念作为主要工具，我们证明所有可在$ t_ \ infty $中定义的组都是（by-by-abelian）by-ealgebraic，特别是回答了Granger的问题。我们得出的结论是，$ t_ \ infty $中可定义的每个无限字段对于矢量空间标量的字段一定是同构的。我们在$ t_ \ infty $中获得了良好行为的其他后果，例如任何可定义的集合中的每个通用类型都是可定义的类型；每组都是扩展基础；每个可定义的组都有一个可定义的连接组件。 我们还考虑了带有非排定双线性形式或非对称的对称阳性双线性双线性形式的真实闭合场上的矢量空间的理论。 Using the same construction as in the case of $T_\infty$, we define a dimension on sets definable in $T^{RCF}_\infty$, and using it we prove analogous results about definable groups and fields: every group definable in $T^{RCF}_{\infty}$ is (semialgebraic-by-abelian)-by-semialgebraic (in特别是（by-by-by-lie） - 在$ t^{rcf} _ {\ infty} $中可以定义的每个字段在标量领域都可以定义，因此它是真正的封闭或代数封闭的。
We study definable sets, groups, and fields in the theory $T_\infty$ of infinite-dimensional vector spaces over an algebraically closed field equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define an ($\mathbb{N}\times \mathbb{Z},\leq_{lex}$)-valued dimension on definable sets in $T_\infty$ enjoying many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all groups definable in $T_\infty$ are (algebraic-by-abelian)-by-algebraic, which, in particular, answers a question of Granger. We conclude that every infinite field definable in $T_\infty$ is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension in $T_\infty$, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component. We also consider the theory $T^{RCF}_\infty$ of vector spaces over a real closed field equipped with a nondegenerate alternating bilinear form or a nondegenerate symmetric positive-definite bilinear form. Using the same construction as in the case of $T_\infty$, we define a dimension on sets definable in $T^{RCF}_\infty$, and using it we prove analogous results about definable groups and fields: every group definable in $T^{RCF}_{\infty}$ is (semialgebraic-by-abelian)-by-semialgebraic (in particular, it is (Lie-by-abelian)-by-Lie), and every field definable in $T^{RCF}_{\infty}$ is definable in the field of scalars, hence it is either real closed or algebraically closed.
