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• W2057500258 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." @default.
• W2057500258 created "2016-06-24" @default.
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• W2057500258 date "2003-12-01" @default.
• W2057500258 modified "2023-10-06" @default.
• W2057500258 title "Cancellative residuated lattices" @default.
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• W2057500258 doi "https://doi.org/10.1007/s00012-003-1822-4" @default.
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