The L-algebras related to prime spectra of Bézout domains and abelian ℓ-groups

The ℓ-spectrum problem asks for a topological characterization of the prime spectrum of a Bézout domain (equivalently, the inverse prime spectrum of an abelian ℓ-group). While a general solution is out of reach, the analogous problem for the maximal spectrum of a Bézout domain was solved in a previous article. An analysis of the ℓ-spectrum problem by means of L-algebras is given. If the prime spectrum is an Esakia space, the known explicit solutions will be compared and related to a finitely additive measure that connects two fundamental classes of L-algebras. The abelian ℓ-groups constructed by several authors from an Esakia space are shown to be structure groups of L-algebras. The L-algebraic method is then extended to more general prime spectra, which leads to a new sufficient criterion for spectral spaces to be representable as prime spectra of Bézout domains.