Real spectrum versus $\ell$-spectrum via Brumfiel spectrum

TL;DR

This paper investigates the relationship between real spectra and ℓ-spectra, showing embeddings, limitations, and counterexamples, and solving a problem posed by Mellor and Tressl in spectral space theory.

Contribution

It establishes new embedding results and limitations between real spectra and ℓ-spectra, and provides counterexamples and solutions to existing open problems.

Findings

Every real spectrum can be embedded into an ℓ-spectrum.

Not all real spectra are ℓ-spectra.

A spectral subspace of a real spectrum may not be a real spectrum.

Abstract

It is well known that the real spectrum of any commutative unital ring, and the ${\ell}$-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 ${\ell}$-spectrum. (2) Not every real spectrum is an ${\ell}$-spectrum. (3) A spectral subspace of a real spectrum may not be a real spectrum. (4) Not every ${\ell}$-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 ${\ell}$-spectrum. The commutative unital rings and Abelian lattice-ordered groups in (2), (3), (4) all have cardinality $\aleph 1 $, while the spectral space of (5) has a basis of cardinality $\aleph 2$. Moreover, (3) solves a problem by Mellor and Tressl.