Über additive und isotone Funktionale auf geordneten Gruppen
In this paper, we use filters of an EQ-algebra E to induce a uniform structure (E, 𝓚), and then the part 𝓚 induce a uniform topology 𝒯 in E. We prove that the pair (E, 𝒯) is a topological EQ-algebra, and some properties of (E, 𝒯) are investigated. In particular, we show that (E, 𝒯) is a first-countable, zero-dimensional, disconnected and completely regular space. Finally, by using convergence of nets, the convergence of topological EQ-algebras is obtained.
Usually, an abelian -group, even an archimedean -group, has a relatively large infinity of distinct -closures. Here, we find a reasonably large class with unique and perfectly describable -closure, the class of archimedean -groups with weak unit which are “-convex”. ( is the group of rationals.) Any is -convex and its unique -closure is the Alexandroff algebra of functions on defined from the clopen sets; this is sometimes .
It is proved that a radical class of lattice-ordered groups has exactly one cover if and only if it is an intersection of some -complement radical class and the big atom over .