## Sequential convergences in distributive lattices.(English)Zbl 0818.06009

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).

### 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
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