Cancellative residuated lattices

Abstract.

Cancellative residuated lattices are natural generalizations of lattice-ordered groups ($\mathcal{l}$-groups). Although cancellative monoids are defined by quasi-equations, the class $\mathcal{CanRL}$ of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of $\mathcal{CanRL}$ 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 $\mathcal{l}$-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 $\mathcal{CanRL}$.

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 $\mathcal{l}$-groups, hence the latter form a variety, denoted by $\mathcal{LG}^-$. Furthermore we prove that the map that sends a subvariety of $\mathcal{l}$-groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of $\mathcal{LG}$ to the lattice of subvarieties of $\mathcal{LG}^-$. 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.