In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend that result to ℓ-spectra of vector lattices over any countable totally ordered division ring $\mathbb{k}$. Combining those methods with Baro's Normal Triangulation Theorem, we obtain the following result:

Theorem

For every countable formally real field $\mathbb{k}$, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital $\mathbb{k}$-algebra.

The countability assumption on $\mathbb{k}$ is necessary: there exists a second countable, completely normal spectral space that cannot be embedded, as a spectral subspace, into either the ℓ-spectrum of any right vector lattice over an uncountable directed partially ordered division ring, or the real spectrum of any commutative unital algebra over an uncountable field.

中文翻译：

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代数和向量格在可数域上的实谱和 ℓ-谱

在前面的文章中，我们建立了每秒可数的，完全正常的频谱空间是同胚于ℓ一些阿贝尔的-spectrum ℓ -group。我们将该结果扩展到任何可数全序划分环上的向量格的ℓ -谱$\mathbb{克}$. 将这些方法与 Baro's Normal Triangulation Theorem 相结合，我们得到以下结果：

定理

对于每一个可数的形式实域 $\mathbb{克}$, 每秒可数的、完全正规的谱空间同胚于某个交换单位的实谱 $\mathbb{克}$-代数。

可数性假设 $\mathbb{克}$是必要的：存在第二个可数的、完全正规的谱空间，它不能作为谱子空间嵌入到不可数有向偏序划分环上的任何右向量格的ℓ -谱，或任何可交换的实谱中。不可数域上的单位代数。