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 $(\omega_1+1)\times(\omega+1)$.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2007
- DOI:
- 10.48550/arXiv.0705.1824
- arXiv:
- arXiv:0705.1824
- Bibcode:
- 2007arXiv0705.1824B
- Keywords:
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- Mathematics - General Topology;
- 54G12;
- 06E05;
- 06A06
- E-Print:
- 19 pages