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Real spectrum versus {\ell}-spectrum via brumfiel spectrum. (arXiv:1706.09802v1 [math.RA])

来源于:arXiv
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 (4) has a basis of car-dinality $\aleph 2. Moreover, (3) solves a problem by Mellor and Tressl. 查看全文>>