  We prove that an inverse-free equation is valid in the variety LG of
lattice-ordered groups (l-groups) if and only if it is valid in the variety DLM
of distributive lattice-ordered monoids (distributive l-monoids). This
contrasts with the fact that, as proved by Repnitskii, there exist inverse-free
equations that are valid in all Abelian l-groups but not in all commutative
distributive l-monoids, and, as we prove here, there exist inverse-free
equations that hold in all totally ordered groups but not in all totally
ordered monoids. We also prove that DLM has the finite model property and a
decidable equational theory, establish a correspondence between the validity of
equations in DLM and the existence of certain right orders on free monoids, and
provide an effective method for reducing the validity of equations in LG to the
validity of equations in DLM.
