Cancellative Residuated Lattices

Cancellative residuated lattices are a natural generalization of lattice-ordered groups (`-groups). Although cancellative monoids are defined by quasi-equations, the class CanRL of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of CanRL that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to `-groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of CanRL. We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of `-groups, hence the latter form a variety, denoted by LG. Furthermore we prove that the map that sends a subvariety of `-groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of LG to the lattice of subvarieties of LG. Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzie’s characterization of categorically equivalent varieties.