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.