Free Boolean algebras over unions of two well orderings |
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Authors: | Robert Bonnet Latifa Faouzi Wies?aw Kubi? |
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Institution: | a Laboratoire de Mathématiques, Université de Savoie, Le Bourget-du-Lac, France b U.F.R. Mathématiques Discrètes et Applications, F.S.T., Université Sidi Mohamed Ben Abdellah, Fès, Morocco c Institute of Mathematics of the Academy of Sciences of the Czech Republic, Czech Republic d Department of Mathematics, Jan Kochanowski University, Kielce, Poland |
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Abstract: | Given a partially ordered set P there exists the most general Boolean algebra 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 κ2 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), showing that there are only ℵ1 many types. In contrast with the last result, we show that there are ℵ12 topological types of closed subsets of the Tikhonov plank (ω1+1)×(ω+1). |
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Keywords: | primary 03G05 06A06 secondary 06E05 08A05 54G12 |
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