Projective objects in the categories of abelian ℓ-groups and MV-algebras

Abstract

We provide a description of free MV-algebras as subalgebras of intervals of free Abelian l-groups. This description shows that the unital Abelian l-group corresponding to a free MV-algebra is projective as an Abelian l-group. More generally, we prove that this is still the case for projective MV-algebras. Using a construction similar to the one for MV-algebras, we show that the free algebras in the variety of negative cones of Abelian l-groups are subalgebras of negative cones of free Abelian l-groups. This allows us to prove a Baker–Beynon-type theorem for finitely generated free algebras in the variety of negative cones of Abelian l-groups. These results are specific cases of a more general situation, which is the subject of the last section of this paper.