Filter exhaustiveness and filter limit theorems for $k$-triangular lattice group-valued set functions

We give some limit theorems for sequences of lattice group-valuedk-triangular set functions,in the setting of filter convergence, and some results about their equivalence. We use the toolof filter exhaustiveness to get uniform (s)-boundedness, uniform continuity and uniform regular-ity of a suitable subsequence of the given sequence, whose indexes belong to the involved filter. Furthermore we pose some open problems.