Relatively Uniformly Complete Φ-Algebras
William Alan Feldman · 2020
The relative uniform convergence for a Φ-algebra (an Archimedean lattice-ordered algebra with an identity which is a weak order unit) is studied in this chapter with the goal of describing its completion. The chapter reviews some pertinent facts about convergence structures limiting ourselves to f-algebras, although the extensions to other cases are evident. It introduces a class of Φ-algebras that will prove important for this study. The chapter defines a new convergence structure for a function algebra V. It relates the ı-convergence structure to the relative uniform convergence structure. The function algebra V is said to be relatively uniformly complete (i.e., complete in the relatively uniform convergence structure p) if each relatively uniformly Cauchy net (filter) converges relatively uniformly to an element in V.