Real Spectrum Versus ℓ-Spectrum via Brumfiel Spectrum

It is well known that the real spectrum of any commutative unital ring, and ℓ-spectrum Abelian lattice-ordered group with order-unit, are all completely normal spectral spaces. We complete existing list containments non-containments between associated spaces their subspaces, by proving following results: (1) Every can be embedded, as a subspace, into some ℓ-spectrum. (2) Not every an (3) A subspace may not spectrum. (4) The rings groups constructed in Eqs. 2, 3, 4 have cardinality ℵ1. Moreover, Eq. 3 solves problem stated 2012 Mellor Tressl.