A continuous $3$-distributive frame that is not ω-distributive
Abstract
We give a negative answer to the question, posed by Erné, whether every $3$-distributive lattice is ω-distributive. More precisely, we exhibit a continuous frame that is κ-distributive for every integer κ≥ 2, but is not a wide coframe. The frame is the open-set lattice of a compact, locally compact, countably based T0 topological meet-semilattice, obtained from Lawson's construction in the logarithmic form described by Goubault-Larrecq. The failure of ω-distributivity is witnessed by an explicit matrix with countably many nonempty finite rows: all row joins are the same nonzero element, whereas every choice of one entry from each row has meet zero. The same space answers negatively Erné's accompanying question whether every $4$-web space is a wide web space. All properties of the construction needed for these conclusions are proved directly.
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BibTeX
@misc{luan2026continuous,
title = {{A continuous $3$-distributive frame that is not $ω$-distributive}},
author = {Wei Luan and Qingguo Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11671},
eprint = {2609.11671},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11671},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A continuous $3$-distributive frame that is not ω-distributive
AU - Wei Luan
AU - Qingguo Li
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11671
UR - https://arxiv.org/abs/2609.11671
AB - We give a negative answer to the question, posed by Erné, whether every $3$-distributive lattice is ω-distributive. More precisely, we exhibit a continuous frame that is κ-distributive for every integer κ≥ 2, but is not a wide coframe. The frame is the open-set lattice of a compact, locally compact, countably based T_0 topological meet-semilattice, obtained from Lawson's construction in the logarithmic form described by Goubault-Larrecq. The failure of ω-distributivity is witnessed by an explicit matrix with countably many nonempty finite rows: all row joins are the same nonzero element, whereas every choice of one entry from each row has meet zero. The same space answers negatively Erné's accompanying question whether every $4$-web space is a wide web space. All properties of the construction needed for these conclusions are proved directly.
ER -