Cancellative residuated lattices

Abstract. Cancellative residuated lattices are a natural general-ization of lattice-ordered groups (`-groups). Although cancellative monoids are de¯ned by quasi-equations, the class CanRL of can-cellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of CanRL that cover the triv-ial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples show-ing 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 cancella-tive 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 de¯ne 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 brie°y discuss these results in the context of R. McKenzie's characterization of categorically equivalent varieties. 1