Relatively uniform convergences in archimedean lattice ordered groups

Abstract

ABSTRACT. For an archimedean lattice ordered group G let G d and G ∧ be the divisible hull or the Dedekind completion of G, respectively. Put G d∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ -radical class; we apply this notion by investigating relative uniform convergences.

Weak relatively uniform convergences on abelian lattice ordered groups

“…The Cauchy completeness of G with respect to α(M )-convergence is investigated. Some results of the paper [14] are extended. We show that if G has the diagonal property, then the system of all α(M )-closed convex -subgroups of G is a complete Browerian lattice.…”

“…the monographs [2], [15], [17]) and in the theory of archimedean lattice ordered groups (cf. the papers [1], [3], [5], [6], [7], [14], [16]). Related notions for M V -algebras were studied in [4].…”

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

“…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 .…”

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.…”