On filter alpha-convergence and exhaustiveness of function nets in lattice groups and applications

We consider (strong uniform) continuity of the limit of a pointwise convergent net of lattice group-valued functions, (strong weak) exhaustiveness and (strong) alpha convergence with respect to a pair of filters, which in the setting of nets are more natural than the corresponding notions formulated with respect to a single filter. Some comparison results are given between such concepts, in connection with suitable properties of filters. Moreover, some modes of filter (strong uniform) continuity for lattice group-valued functions are investigated, giving some characterization. As an application, we get some Ascoli-type theorem in an abstract setting, extending earlier results to the context of filter alpha-convergence.