Real spectra and ℓ-spectra of algebras and vector lattices over countable fields

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 result ℓ-spectra vector lattices over any countable totally ordered division ring k. Combining those methods with Baro's Normal Triangulation Theorem, obtain following result: TheoremFor formally real field k, spectrum commutative unital k-algebra. The countability assumption on k necessary: there exists a cannot be embedded, as subspace, into either right lattice uncountable directed partially ring, or algebra field. Theorem For