Locally solid convergences and order continuity of positive operators

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