We study \({\mathfrak {u}}\) O-convergence on infinitely distributive lattices, extending key properties known from Riesz spaces. We show that order continuity of \({\mathfrak {u}}\) O-convergence characterizes infinite distributivity. We examine O-adherence and \({\mathfrak {u}}\) O-adherence of sublattices and ideals, proving that the \({\mathfrak {u}}\) O- and O-closures of a sublattice coincide and form a sublattice, and that the first \({\mathfrak {u}}\) O-adherence of an ideal is an O-closed ideal. We also analyze the Dedekind-MacNeille completion of a sublattice Y within that of a lattice L, identifying conditions (A) and (B) under which the completion of Y embeds regularly in that of L. In this case, we show that the first \({\mathfrak {u}}\) O-adherence of Y covers its O-closure.
