双线性空间在固定域上的简单不稳定
Bilinear spaces over a fixed field are simple unstable
AI总结：
研究双线性形式向量空间的模型论，针对无限域需不同逻辑框架，通过范畴论和正逻辑方法，证明理论简单不稳定，并展示双线性空间在可数域上ω-范畴性。
摘要：
我们研究固定域上带有双线性形式的向量空间的模型论。对于有限域，这在全一阶逻辑的经典框架中已得以完成。对于无限域，我们需要不同的逻辑框架。首先采用范畴论方法，要求极小的设前提。我们证明线性独立形成一个简单的不稳定独立关系。进一步工作显示我们也可以在正逻辑框架中工作，该框架比范畴论方法更强大，更接近全一阶逻辑的经典框架。我们完全刻画了出现的正理论的完全闭模型。利用之前的独立关系，我们得出该理论是简单不稳定的，即除法有局部特征但存在许多不同类型。我们还提供了通常称为全一阶逻辑中ω-范畴理论的Ryll-Nardzewski定理的正版本，从而得出双线性空间在可数域上是ω-范畴的结论。
英文摘要：
We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical frameworks. First we take a category-theoretic approach, which requires very little set-up. We show that linear independence forms a simple unstable independence relation. With some more work we then show that we can also work in the framework of positive logic, which is much more powerful than the category-theoretic approach and much closer to the classical framework of full first-order logic. We fully characterise the existentially closed models of the arising positive theory. Using the independence relation from before we conclude that the theory is simple unstable, in the sense that dividing has local character but there are many distinct types. We also provide positive version of what is commonly known as the Ryll-Nardzewski theorem for $ω$-categorical theories in full first-order logic, from which we conclude that bilinear spaces over a countable field are $ω$-categorical.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
