‎Completeness for Saturated $\mathsf{L}$-Quasi-Uniform Limit Spaces

‎We define and study two completeness notions for saturated $\mathsf{L}$-quasi-uniform limit spaces‎. ‎The one‎, ‎that we term Lawvere completeness‎, ‎is defined using the concept of promodule and lends a lax algebraic interpretation of completeness also for saturated $\mathsf{L}$-quasi-uniform limit spaces‎. ‎The other‎, ‎termed Cauchy completeness‎, ‎is defined using saturated Cauchy pair prefilters‎. ‎We show that both concepts coincide with related notions in the case of saturated $\mathsf{L}$-quasi-uniform spaces and that also for saturated $\mathsf{L}$-quasi-uniform limit spaces‎, ‎both completeness notions are equivalent‎.