Strictly join irreducible varieties of residuated lattices

We study (strictly) join irreducible varieties in the lattice of subvarieties of residuated lattices. We explore the connections with well-connected algebras and suitable generalizations, focusing in particular on representable varieties. Moreover, we find weakened notions of Halldén completeness that characterize join irreducibility. We characterize strictly join irreducible varieties of basic hoops and use the generalized rotation construction to find strictly join irreducible varieties in subvarieties of $\mathsf{MTL}$-algebras. We also obtain some general results about linear varieties of residuated lattices, with a particular focus on representable varieties, and a characterization for linear varieties of basic hoops.