| Abstract:
It is possible to attach a 'spectrum' to several algebraic structures:
commutative rings with identity, bounded distributives lattices,
commutative semirings with identity, commutative \(C^{*}\)-algebras, commutative
monoids, abelian \(\ell\)-groups, MV-algebras, continuous lattices,
Zariski-Riemann spaces,... In all these cases, the spectrum turns out to
be a spectral topological space, i.e., a sober compact space in which the
intersection of any two compact open sets is compact, and the compact opens
forms a basis for the topology. On some other cases, for instance, for
commutative rings without identity or noncommutative rings, one gets a
spectrum that is a little less: it is always a sober space, but sometimes
compactness is missing, or the intersection of two compact open sets is not
necessarily compact. In this talk we will investigate the reason of this
'ubiquity of spectral spaces', giving the proper setting for this kind of
questions: properly defined multiplicative lattices. We will present work
in progress with Carmelo Antonio Finocchiaro and George Janelidze.
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