Minimal Varieties of Representable Commutative Residuated Lattices

We solve several open problems on the cardinality of atoms in the subvariety lattice of residuated lattices and FL-algebras [4, Problems 17–19, pp. 437]. Namely, we prove that the subvariety lattice of residuated lattices contains continuum many 4-potent commutative representable atoms. Analogous results apply also to atoms in the subvariety lattice of FLi-algebras and FLo-algebras. On the other hand, we show that the subvariety lattice of residuated lattices contains only five 3-potent commutative representable atoms and two integral commutative representable atoms. Inspired by the construction of atoms, we are also able to prove that the variety of integral commutative representable residuated lattices is generated by its 1-generated finite members.