论文标题
固定字段上的双线性空间很简单不稳定
Bilinear spaces over a fixed field are simple unstable
论文作者
论文摘要
我们在固定场上研究具有双线性形式的向量空间的模型理论。对于有限字段，这可以并且已经在完整的一阶逻辑的经典框架中完成。对于无限字段，我们需要不同的逻辑框架。首先，我们采用类别理论方法，这几乎不需要设置。我们表明，线性独立性形成了一个简单的不稳定独立关系。然后，我们还可以证明，我们还可以在积极逻辑的框架中工作，这比类别理论方法强大得多，并且更接近完整的一阶逻辑的经典框架。我们充分表征了阳性理论的存在封闭的模型。在我们得出结论之前，使用独立关系，理论是简单的不稳定的，从某种意义上说，分裂具有本地特征，但是有许多不同的类型。我们还提供了普遍称为Ryll-Nardzewski定理的正面版本，以$ω$ - 分类理论完整的一阶逻辑，从中我们可以得出结论，在该逻辑上，在可数字段上的双线性空间为$ω$ - 类别。
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.
