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

DOI10.1007/S10468-021-10088-0arXiv1706.09802MaRDI QIDQ6288444

Author name not available (Why is that?)

Publication date: 29 June 2017

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 $aleph1$, while the spectral space of (5) has a basis of cardinality $aleph2$. Moreover, (3) solves a problem by Mellor and Tressl.

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