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Idempotent Ultrafilters and Closed Images of Complete Semilattices

  • Alexander V. Osipov
  • Konstantin Kazachenko
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Abstract

We prove that every continuous homomorphism from a complete topologized semilattice to a Hausdorff semitopological semilattice has closed image, answering a question of Banakh and Bardyla. The main step is the convergence of idempotent ultrafilters on complete semitopological semilattices. We also prove that, for every complete subsemilattice X of a Hausdorff semitopological semilattice Y, the natural order of X is closed in Y× Y. In particular, the natural order of every complete Hausdorff semitopological semilattice X is closed in X× X.

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Cite

BibTeXRIS
BibTeX
@misc{osipov2026idempotent,
  title = {{Idempotent Ultrafilters and Closed Images of Complete Semilattices}},
  author = {Alexander V. Osipov and Konstantin Kazachenko},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01545},
  eprint = {2611.01545},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01545},
}
RIS
TY  - UNPB
TI  - Idempotent Ultrafilters and Closed Images of Complete Semilattices
AU  - Alexander V. Osipov
AU  - Konstantin Kazachenko
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01545
UR  - https://arxiv.org/abs/2611.01545
AB  - We prove that every continuous homomorphism from a complete topologized semilattice to a Hausdorff semitopological semilattice has closed image, answering a question of Banakh and Bardyla. The main step is the convergence of idempotent ultrafilters on complete semitopological semilattices. We also prove that, for every complete subsemilattice X of a Hausdorff semitopological semilattice Y, the natural order of X is closed in Y× Y. In particular, the natural order of every complete Hausdorff semitopological semilattice X is closed in X× X.
ER  -