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Compactly generated De Morgan lattices, basic algebras and effect algebras. (English) Zbl 1202.06007

Summary: We prove that a De Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on \(L\) generated by some separating function family on \(L\). Moreover, if \(L\) is complete then \(L\) is (o)-topological. Further, if a basic algebra \(L\) (hence a lattice with sectional antitone involutions) is compactly generated then \(L\) is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.

MSC:

06C15 Complemented lattices, orthocomplemented lattices and posets
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
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