Near Regularly-Prüfer Rings

In the present article, we construct the theory of near Boolean families of valuation rings which extends the corresponding theory for Boolean families. We establish the fact that the local-global principle (LGP) for such families is effectively elementary. We indicate sufficient conditions for 1-embeddability of holomorphy rings of near Boolean families that possess the LGP property. By way of application, we prove that the elementary theory of the ring of integrably totally $p$-adic integers and the elementary theory of the class of all closed subrings of almost all algebraic numbers are decidable.