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Free Boolean algebras over unions of two well orderings
Authors:Robert Bonnet  Latifa Faouzi  Wies?aw Kubi?
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
Abstract:Given a partially ordered set P there exists the most general Boolean algebra View the MathML source 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=P0P1, 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 1 many types. In contrast with the last result, we show that there are 12 topological types of closed subsets of the Tikhonov plank (ω1+1)×(ω+1).
Keywords:primary  03G05  06A06  secondary  06E05  08A05  54G12
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