Interpolation in Hájek's basic logic

We exhaustively classify varieties of BL-algebras with the amalgamation property, showing that there are only countably many of them and solving an open problem of Montagna. As a consequence of this classification, we obtain a complete description of which axiomatic extensions of Hájek's basic fuzzy logic BL have the deductive interpolation property. Along the way, we provide similar classifications of varieties of basic hoops with the amalgamation property and axiomatic extensions of the negation-free fragment of BL with the deductive interpolation property.