Lattice-ordered abelian groups, or abelian l-groups in what follows, are categorically equivalent to two classes of 0-bounded hoops that are relevant in the realm of the equivalent algebraic semantics of many-valued logics: liftings of cancellative hoops and perfect MV-algebras. The former generate the variety of product algebras, and the latter the subvariety of MV-algebras generated by perfect MV-algebras, that we shall call DLMV. In this seminar we focus on these two varieties and their relation to the structures obtained by forgetting the falsum constant 0, i.e., product hoops and DLW-hoops.

A first main result is a characterization of the free algebras over an arbitrary set of generators in the two varieties of product and DLW-hoops; the latter are obtained as particular subreducts of the corresponding free algebras in the 0-bounded varieties. More precisely, we obtain a representation in terms of weak Boolean products of which we characterize the factors. This kind of description for 0-bounded residuated lattices is present in the literature, but we are not aware of analogous results for varieties of residuated structures with just the constant 1.

We observe that in a variety that is the equivalent algebraic semantics of a logic, free (finitely generated) algebras are isomorphic to the Lindenbaum-Tarski algebras of formulas of the logic; thus their study is important from both the perspective of algebra and logic.