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Jumpy lattices. (English) Zbl 1245.06018
Chajda, I. (ed.) et al., Proceedings of the 79th workshop on general algebra “79. Arbeitstagung Allgemeine Algebra”, 25th conference of young algebraists, Palack√Ĺ University Olomouc, Olomouc, Czech Republic, February 12–14, 2010. Klagenfurt: Verlag Johannes Heyn (ISBN 978-3-7084-0407-3/pbk). Contributions to General Algebra 19, 159-171 (2010).
Author’s abstract: “We say that a lattice \(L\) contains a jump if there are \(a, b \in L\) so that \(a\prec b\). \(L\) is said to be jumpy if all of its nontrivial homomorphic images have a jump. If \(L\) is Boolean, then every covering relationship \(a\prec b\) in \(L\) is associated with its atom \(b\backslash a\), so that a Boolean algebra is jumpy if and only if it is superatomic. We show that an arbitrary bounded distributive lattice is jumpy if and only if its Boolean reflection is superatomic if and only if its Priestley space is scattered. The class of jumpy lattices is closed under sublattices, homomorphic images (quotient lattices), and weak direct products.
A classical result of Day characterizes superatomic Boolean algebras as those which admit a reflection into complete Boolean lattices. The corresponding property for bounded distributive lattices would be a reflection into doubly algebraic lattices. We show that jumpy lattices admit such a reflection, but that this property does not characterize them.”
For the entire collection see [Zbl 1201.08001].
06D05 Structure and representation theory of distributive lattices
06E15 Stone spaces (Boolean spaces) and related structures