Real spectrum versus ℓ-spectrum via Brumfiel spectrum
Résumé
It is well known that the real spectrum of any commutative unital ring, and the ℓ-spectrum of any Abelian lattice-ordered group with order-unit, are all completely normal spectral spaces. We prove the following results: (1) Every real spectrum can be embedded, as a spectral subspace, into some ℓ-spectrum. (2) Not every real spectrum is an ℓ-spectrum. (3) A spectral subspace of a real spectrum may not be a real spectrum. (4) Not every ℓ-spectrum can be embedded, as a spectral subspace, into a real spectrum. (5) There exists a completely normal spectral space which cannot be embedded , as a spectral subspace, into any ℓ-spectrum. The commutative unital rings and Abelian lattice-ordered groups in (2), (3), (4) all have cardinality ℵ 1 , while the spectral space of (5) has a basis of cardinality ℵ 2. Moreover, (3) solves a problem by Mellor and Tressl.
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