Cancellative residuated lattices

Cancellative residuated lattices are natural generalizations of lattice-orderedgroups (-groups). Although cancellative monoids are defined by quasi-equations, the class of cancellative residuated lattices is a variety. We prove that there are only twocommutative subvarieties of that cover the trivial variety, namely the varietiesgenerated by the integers and the negative integers (with zero). We also construct examplesshowing that in contrast to -groups, the lattice reducts of cancellative residuated latticesneed not be distributive. In fact we prove that every lattice can be embedded in thelattice reduct of a cancellative residuated lattice. Moreover, we show that there exists anorder-preserving injection of the lattice of all lattice varieties into the subvariety lattice of.We define generalized MV-algebras and generalized BL-algebras and prove that thecancellative integral members of these varieties are precisely the negative cones of -groups, hence the latter form a variety, denoted by . Furthermore we prove that the map that sends a subvariety of -groups to the corresponding class of negative cones is a latticeisomorphism from the lattice of subvarieties of to the lattice of subvarieties of .Finally, we show how to translate equational bases between corresponding subvarieties, andbriefly discuss these results in the context of R. McKenzies characterization of categoricallyequivalent varieties.