Real spectrum versus ℓ-spectrum via Brumfiel spectrum

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: distributive; lattice; closed map; real spectrum; representable; l-spectrum; spectrum; spectral space; completely normal; radical; prime; ideal; real-closed; f-ring; l-group; Brumfiel spectrum; convex map; condensate; Stone duality; sober

Domaines: Mathématiques générales [math.GM]

Friedrich Wehrung. Real spectrum versus ℓ-spectrum via Brumfiel spectrum. 2017. ⟨hal-01550450v2⟩