Abstract: For an abelian lattice ordered group $G$ let $\conv G$ 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 $\alpha _{nd}$ 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 $\conv G$. In the present paper we prove that the join $\alpha _{nd}\vee \alpha $ does exist in $\conv G$.
Keywords: convergence $\ell$-group, nearly disjoint sequence, strong convergence
Classification (MSC2000): 06F20, 22C05
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