Free Boolean algebras over unions of two well orderings

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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