Sequential convergences in distributive lattices. (English) Zbl 0818.06009

Jakubík, Ján

Math. Bohem. 119, No. 3, 245-254 (1994).

Let \(L\) be a distributive lattice and let \(\text{Conv }L\) denote the partially ordered set of all sequential convergences on \(L\). The author has found a new necessary and sufficient condition for \(\text{Conv }L\) to be a complete lattice. This condition is applied to prove that if \(L\) is \((\aleph_ 0,2)\)-distributive, then \(\text{Conv } L\) is a complete lattice. (The analogous results for \(\ell\)-groups and for Boolean algebras were established by the author in Czech. Math. J. 40, 453-458 (1990; Zbl 0731.06010).) Furthermore, in the third section of the paper there is shown that if \(L\) is a direct product of \(L_ i\) \((i\in I)\) and \(\alpha\in \text{Conv } L\), \(\alpha= \prod_{i\in I} \alpha_ i\), \(\alpha_ i\in \text{Conv } L_ i\), then \(\alpha\) is a maximal element of \(\text{Conv } L\) if and only if \(\alpha_ i\) is a maximal element of \(\text{Conv } L_ i\) for each \(i\in I\). (The last result holds also in the case of \(\ell\)-groups [see the reviewer and the author, Czech. Math. J. 39, 631-640 (1989; Zbl 0703.06011).

Reviewer: M.Harminc (Košice)

MSC:

06D99	Distributive lattices
06B30	Topological lattices
22A26	Topological semilattices, lattices and applications

Keywords:

sequential convergence; distributive lattice

Citations:

Zbl 0731.06010; Zbl 0703.06011