Nearly disjoint sequences in convergence \(\ell \)-groups. (English) Zbl 0967.06013

Jakubík, Ján

Math. Bohem. 125, No. 2, 139-144 (2000).

Summary: For an abelian lattice-ordered group \(G\) let \(\text{conv} G\) be the system of all compatible convergences on \(G\); this system is a meet semilattice but in general it fails to be a lattice. Let \(\alpha _{\text{nd}}\) be the convergence on \(G\) which is generated by the set of all nearly disjoint sequences in \(G\), and let \(\alpha \) be any element of \(\text{conv} G\). In the present paper we prove that the join \(\alpha _{\text{nd}}\vee \alpha \) does exist in \(\text{conv} G\).

MSC:

06F20	Ordered abelian groups, Riesz groups, ordered linear spaces
22C05	Compact groups

Keywords:

convergence \(\ell \)-group; nearly disjoint sequence; strong convergence