两个阱序并上的自由布尔代数
Free Boolean algebras over unions of two well orderings
摘要：
给定一个偏序集$P$，存在以$P$为生成集的最一般布尔代数$F(P)$，称为$P$上的{ \it自由布尔代数}.研究了$P=P_0\cup P_1$型偏序集上的自由布尔代数，其中$P_0,P_1$是良序.我们称它们为{ \it近序代数}。讨论了Maurice Pouzet的一个问题，证明了对每一个不可数基数$κ$，都存在基数$κ$的$2^κ$两两非同构的近序代数.在拓扑上，偏序集上的自由布尔代数对应于紧0维分配格。在这种情况下，我们对积$(ω_1+1)\times(ω_1+1)$的所有闭子晶格进行分类，从而表明只有$\aleph_1$，其中许多。与上一个结果相比，我们证明了Tikhonov Plank $(ω_1+1)\times(ω+1)$的闭子集存在$2^{\aleph_1}$拓扑类型.
英文摘要：
Given a partially ordered set $P$ there exists the most general Boolean algebra $F(P)$ which contains $P$ as a generating set, called the {\it free Boolean algebra} over $P$. We study free Boolean algebras over posets of the form $P=P_0\cup P_1$, where $P_0,P_1$ are well orderings. We call them {\it nearly ordinal algebras}. Answering a question of Maurice Pouzet, we show that for every uncountable cardinal $κ$ there are $2^κ$ pairwise non-isomorphic nearly ordinal algebras of cardinality $κ$. Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product $(ω_1+1)\times(ω_1+1)$, thus showing that there are only $\aleph_1$ many of them. In contrast with the last result, we show that there are $2^{\aleph_1}$ topological types of closed subsets of the Tikhonov plank $(ω_1+1)\times(ω+1)$.
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