一个连续$3$-分配格但不是$\omega$-分配格

中文摘要 AI 辅助

我们否定了Erné提出的问题：是否每个$3$-分配格都是$\omega$-分配格。更精确地说，我们构造了一个连续框架，它对每个整数$\kappa\geq 2$都是$\kappa$-分配的，但不是宽余框架。该框架是紧的、局部紧的、可数基的$T_0$拓扑交半格的开集格，由Lawson构造以Goubault-Larrecq描述的对数形式得到。$\omega$-分配性的失败由一个显式矩阵见证，该矩阵具有可数多个非空有限行：所有行的并是同一个非零元素，而每行选取一个元素的任何选择之交为零。同一空间也否定了Erné的伴随问题：是否每个$4$-网空间都是宽网空间。构造所需的所有性质都被直接证明。

英文摘要

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 $κ\geq 2$, 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.

发表机构

• Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）
• School of Mathematics, Hunan University（湖南大学数学学院）

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