  Necessary and sufficient conditions are presented for the (first-order)
theory of a universal class of algebraic structures (algebras) to admit a model
completion, extending a characterization provided by Wheeler. For varieties of
algebras that have equationally definable principal congruences and the compact
intersection property, these conditions yield a more elegant characterization
obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski.
Moreover, it is shown that under certain further assumptions on congruence
lattices, the existence of a model completion implies that the variety has
equationally definable principal congruences. This result is then used to
provide necessary and sufficient conditions for the existence of a model
completion for theories of Hamiltonian varieties of pointed residuated
lattices, a broad family of varieties that includes lattice-ordered abelian
groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of
pointed residuated lattices admits a model completion, it must have
equationally definable principal congruences. In particular, the theories of
lattice-ordered abelian groups and MV-algebras do not have a model completion,
as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is
shown that certain varieties of pointed residuated lattices generated by their
linearly ordered members, including lattice-ordered abelian groups and
MV-algebras, can be extended with a binary operation in order to obtain
theories that do have a model completion.
