We study the theory of K-vector spaces with a predicate for the union X of an infinite family of independent subspaces. We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-fold sums of X with itself. If K is finite this is no longer true, but we still have that a natural completion is near-model-complete.

中文翻译：

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具有独立子空间并集的向量空间

我们研究K向量空间的理论，其谓词用于无限独立子空间族的并集X。我们证明，如果K是无限的，那么该理论是完整的，并且允许K向量空间语言中的量词消除，并带有X与其自身的n倍和的谓词。如果K是有限的，则不再正确，但我们仍然认为自然完成是接近模型完成的。