l-Groups and Bézout Domains

We study the relationship between l-groups, Bézout domains, and MV -algebras. Our main motivation and starting point has been the Jaffard-Ohm correspondence between Abelian l-groups and Bézout domains ¶ μ 3 ́ Bézout domains with non-zero unit radical ¶ μ 3 ́ Unital l-groups ¶ μ 3 ́ MV -algebras ©© ©© ©© © HHHHHH ——————————– and Mundici’s equivalence between Abelian unital l-groups and MV -algebras. Using valuation theory, we give a positive answer to Conrad and Dauns’ problem [41] whether a lattice-ordered skew-field with positive squares is linearly ordered. As a counterpart, we prove the existence of directed algebras with negative squares. For an arbitrary l-group, we give some characterizations of l-ideals in terms of absolute values, generalizing a similar result for o-ideals in Riesz spaces [1]. A number of ring-theoretical notions and properties are introduced for lgroups and MV-algebras. Using the correspondence between l-groups and MV algebras, we answer a question of Belluce [9] on prime annihilators in MV algebras in the negative. For an Abelian l-group G, we construct a dense embedding G ↪→ E(G) into a laterally complete Abelian l-group E(G) by means of sheaf theory. In case G is Archimedean, we prove that E(G) is the lateral completion of G, while in general, this seems to be false. As a byproduct, we get a natural and elegant proof of Bernau’s celebrated embedding theorem for Archimedean l-groups. If G is the group of divisibility of a Bézout domain D, we show that Spec(D) is related to a quasi-compact topology on Spec∗(G) which turns out to be the “Hochster dual” (see Section 5.2) of the spectral topology. We study the C-topology [65] on Abelian l-groups and apply it to Bézout domains and MV-algebras. We correct two lemmas of Gusić [65] which leads to a generalization of one of his main results. Finally, we answer a question of Dumitrescu, Lequain, Mott and Zafrullah [50] which shows that the Jaffard-Ohm correspondence does not hold for Abelian almost l-groups.