Varieties generated by unital abelian \(\ell\)-groups (Q401022)

This paper deals with the relationships between lattice-ordered abelian groups (\(l\)-groups for short) and MV-algebras. While Chang's MV-algebras are the algebraic semantics for Łukasiewicz infinite-valued propositional logic, \(l\)-groups provide a modern equational framework for the classical theory of magnitudes. In the theory of unital \(l\)-groups one also takes into account the unit of measurent; as a generalization, positively pointed abelian \(l\)-groups do not require the unit to be an Archimedean element.NEWLINENEWLINE As proved by the present reviewer in his paper [J. Funct. Anal. 65, 15--63 (1986; Zbl 0597.46059)], MV-algebras are categorically equivalent to unital \(l\)-groups. The Archimedean property of the unit in unital \(l\)-groups make their class first-order undefinable.NEWLINENEWLINE On the other hand, the author proves that the subvariety lattice of the variety of positively pointed abelian \(l\)-groups (with the exception of the trivial variety) is isomorphic to the subvariety lattice of MV-algebras. Further (with the exception of the unique atom), each subvariety of positively pointed abelian \(l\)-groups is generated by its unital members.