Three classes of uniform spaces
Egoroff, σ, and convergence properties in some archimedean vector lattices
An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each there are and a ∈ A with λₙaₙ ≤ a for each n; (2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u. The conjunction of these two is called strongly Egoroff. We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent...
Some comments and examples on generation of (hyper-)archimedean -groups and -rings
This paper systematizes some theory concerning the generation of -groups and reduced -rings from substructures. We are particularly concerned with archimedean and hyperarchimedean groups and rings. We discuss the process of adjoining a weak order unit to an -group, or an identity to an -ring and find significant contrasts between these cases. In -groups, hyperarchimedeanness and similar properties fail to pass from generating structures to the structures that they generate, as illustrated by...
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