Abstract
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 complete the existing list of containments and non-containments between the associated spectral spaces and their spectral subspaces, by proving 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. The commutative unital rings and Abelian lattice-ordered groups constructed in Eqs. 2, 3, and 4 all have cardinality ℵ1. Moreover, Eq. 3 solves a problem stated in 2012 by Mellor and Tressl.
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