Abstract
We show that, for a large class of countable order types α, a modular algebraic lattice L contains no chain of type α if and only if K(L), the join-semilattice of compact elements of L, contains neither a chain of type α nor a join-subsemilattice isomorphic to [ω] < ω, the set of finite subsets of ω ordered by inclusion. We give a description of the indivisible members of this class. It includes the order types ω * of the chain of negative integers and η of the chain of rational numbers.
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The second author was supported by INTAS project Universal Algebra and Lattice Theory.
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Chakir, I., Pouzet, M. The Length of Chains in Modular Algebraic Lattices. Order 24, 227–247 (2007). https://doi.org/10.1007/s11083-007-9070-4
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DOI: https://doi.org/10.1007/s11083-007-9070-4