Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions

Abstract

This paper studies sequential order convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified, for compact spaces X, in [ 6 ]. This condition is that every dense cozero set S in X should be C * -embedded in X (that is, all bounded continuous functions on S extend to X ). We call Tychonoff spaces X with this property quasi-F spaces (since they generalize the F -spaces of [ 12 ]). In Section 1, the notion of a completion with respect to sequential order convergence is first described in the setting of a commutative lattice group G.