MV-algebras are algebraic models of Lukasiewicz invinite-valued logic and were introduced by C.C. Chang, who also noticed that the totally-ordered ones are isomorphic to intervals in totally-ordered abelian groups. Mundici extended this correspondence, beyond the totally-ordered case, to MV-algebras and abelian lattice-ordered groups and proved that the associated categories are equivalent. The categorical equivalence was further extended to the non-commutative case (Dvurecenskij) between pseudo-MV-algebras and l-groups with a strong order unit, and to the non-commutative unbouded case (G. - Tsinakis) between generalized MV-algebras and l-groups with a special operator. A useful setting to consider all these structures is that of residuated lattices. We will review the main ideas and outline the associated construction (of embedding a generalized MV-algebra into an l-group).

Varieties (that is, equationally defined classes) of lattice-ordered groups have been well understood for a very long time, and varieties of MV-algebras have been completely understood. But this was not the case for pseudo-MV algebras, which are the ones that are not necessarily commutative.

The exact connection between lattice-ordered groups and pseudo-MV algebras involves the presence in the lattice-ordered group of a unit. That is, an element u such that every element of the group is trapped between powers of the unit. The important thing is that the unit is part of the language. This vastly increases the possible varieties of unital lattice-ordered groups.

The collection of varieties of pseudo-MV algebras is categorically equivalent to the collection of varieties of unital lattice-ordered groups. I will discuss many of the interesting discoveries of these varieties, and raise interesting unsolved questions about them.