  We consider vector lattices endowed with locally solid convergence
structures, which are not necessarily topological. We show that such a
convergence is defined by the convergence to $0$ on the positive cone. Some
results on unbounded modification which were only available in partial cases
are generalized. Order convergence is characterized as the strongest locally
solid convergence in which monotone nets converge to their extremums (if they
exist). We partially characterize sublattices on which the order convergence is
the restriction of the order convergence of the ambient lattice. We show that
homomorphism is order continuous iff it is uo-continuous. Uo convergence is
characterized independently of order convergence. We show that on the space of
continuous function uo convergence is weaker than the compact open convergence
iff the underlying topological space contains a dense locally compact subspace.
For a large class of convergences we prove that a positive operator is order
continuous if and only if its restriction to a dense regular sublattice is
order continuous, and that the closure of a regular sublattice is regular with
the original sublattice being order dense in the closure. We also present an
example of a regular sublattice of a locally solid topological vector lattice
whose closure is not regular.
