Amalgamation in varieties of residuated lattices
Simon Elia Santschi · Open Access CRIS of the University of Bern · 2026
We investigate the amalgamation property (AP) in varieties of residuated lattices. This endeavor is motivated both from the purely algebraic point of view as well as from the logical point of view through the importance of residuated lattices as algebraic semantics for substructural logics. Apart from proving new results, we also survey the state of the art. This thesis may, therefore, also serve as an introduction to the topic for readers acquainted with the theory of residuated lattices. In addition, the thesis contains a number of open problems in each chapter. In the initial two chapters, we recall the necessary preliminaries and then present the important notions and tools needed for our investigation. Our first major result is a proof that the variety of residuated lattices does not have the AP, which resolves a long-standing open problem in the area. After that, we show that there are continuum-many varieties of commutative residuated lattices with the AP. Next, we shift our focus to semilinear varieties of residuated lattices, i.e., varieties generated by their totally ordered members. For the variety of idempotent semilinear residuated lattices we characterize its 60 commutative subvarieties with the AP and show that it has continuum-many non-commutative subvarieties with the AP. Then we prove that the variety of commutative semilinear residuated lattices and many of its subvarieties fail the AP. Finally, we provide a complete characterization of the varieties of BL-algebras and basic hoops with the AP, resolving a problem of Montagna. We conclude with a discussion of possible future directions and a summary of the state of the art. In particular, we provide a table summarizing what is known about the AP and related properties in various varieties of residuated lattices.