‎completeness for saturated $mathsf{l}$-quasi-uniform limit spaces

We define and study two completeness notions for saturated l-quasi-uniform limit spaces. the one, thatwe term lawvere completeness, is defined using the concept of promodule and lends a lax algebraic interpretationof completeness also for saturated l-quasi-uniform limit spaces. the other, termed cauchy completeness, is definedusing saturated cauchy pair prefilters. we show that both concepts coincide with related notions in the caseof saturated l-quasi-uniform spaces and that also for saturated l-quasi-uniform limit spaces, both completenessnotions are equivalent.