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Keywords:
nearly disjoint sequence; strong convergence; convergence $\ell$-group
nearly disjoint sequence; strong convergence; convergence $\ell$-group
Summary:
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 $\conv G$.
References:
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[2] J. Jakubík: Sequential convergences in l-groups without Urysohn's axiom. Czechoslovak Math. J. 42 (1992), 101-116. MR 1152174 | Zbl 0770.06008
[3] J. Jakubík: Disjoint sequences in Boolean algebras. Math. Bohem 123 (1998), 411-418. MR 1667113
[4] E. P. Shimbireva: On the theory of partially ordered groups. Matem. Sbornik 20 (1947), 145-178. (In Russian.) MR 0020558 | Zbl 0029.10301
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