Nearly spectral spaces
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We study some natural generalizations of spectral spaces in the contexts of commutative rings and distributive lattices. We obtain a topological characterization for the spectra of commutative (not necessarily unitary) rings and we find spectral versions for the up-spectral and down-spectral spaces. We show that the duality between distributive lattices and Balbes–Dwinger spaces is the co-equivalence associated with a pair of contravariant right adjoint functors between suitable categories.
KeywordsSpectral space Down-spectral space Up-spectral space Stone duality Prime spectrum Distributive lattice Commutative ring
Mathematics Subject Classification54H10 54F65 54D35
We want to thank the referees for their useful comments.