Cancellative residuated lattices arising on 2-generated submonoids of natural numbers

It is known that there are only two cancellative atoms in the subvariety lattice of residuated lattices, namely the variety of Abelian ?-groups ${\mathcal{CLG}}$ generated by the additive ?-group of integers and the variety ${\mathcal{CLG}^-}$ generated by the negative cone of this ?-group. In this paper we consider all cancellative residuated chains arising on 2-generated submonoids of natural numbers and show that almost all of them generate a cover of ${\mathcal{CLG}^-}$ . This proves that there are infinitely many covers above ${\mathcal{CLG}^-}$ which are commutative, integral, and representable.