Free Boolean algebras over unions of two well orderings

الأصل · EN

Given a partially ordered set P there exists the most general Boolean algebra F(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=P₀∪ P₁, where P₀,P₁ 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 2κ 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 (ω₁+1)×(ω₁+1), thus showing that there are only ₁ many of them. In contrast with the last result, we show that there are 2₁ topological types of closed subsets of the Tikhonov plank (ω₁+1)×(ω+1).