Free Boolean algebras over unions of two well orderings

Abstract

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 $\kappa$ there are $2^{\kappa }$ pairwise non-isomorphic nearly ordinal algebras of cardinality $\kappa$. Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product $({\omega }_1+1)\times ({\omega }_1+1)$, 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 $({\omega }_1+1)\times (\omega +1)$.

arXiv comment19 pages