Free Boolean algebras over unions of two well orderings

Given a partially ordered set P there exists the most general Boolean algebra F ˆ ( P ) which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P = P 0 ∪ P 1 , where P 0 , P 1 are well orderings. We call them 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 ) × ( ω 1 + 1 ) , showing that there are only ℵ 1 many types. In contrast with the last result, we show that there are 2 ℵ 1 topological types of closed subsets of the Tikhonov plank ( ω 1 + 1 ) × ( ω + 1 ) .