On a relative uniform completion of an archimedean lattice ordered group

Abstract

ABSTRACT. The notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups.Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) G ω 1 of G is dealt with. It is known that G ω 1 exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in G ω 1 . Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence.

Cantor extension of an abelian lattice ordered group equipped with a weak relatively uniform convergence

“…The proof is the same as in [5] for the case of ru-convergence in an archimedean lattice ordered group.…”

“…When studying ru-convergence in archimedean lattice ordered groups, analogous results to 5.1 and 5.2 were obtained in [4] and [5] under the assumption that K is a direct factor of G.…”

“…The relatively uniform completion (ru-completion, for short) of an archimedean lattice ordered group has been investigated by Hager and Martinez [8], Ball and Hager [1], Martinez [12],Černák and Lihová [5], and Jakubík andČernák [11]. The ru-completion of an archimedean lattice ordered group G is denoted by G ω 1 .…”

“…The relatively uniform convergence in archimedean vector lattices and in archimedean lattice ordered groups has been studied in several papers; for references, cf.Černák [2] andČernák and Lihová [5].…”

Weak relatively uniform convergences on MV-algebras

“…[17], [21]) and later in archimedean lattice ordered groups (cf. [2], [8], [9], [16], [18]). The notion of a regulator of a convergent sequence is essential in this theory.…”