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Strong compact elements in multiplicative lattices. (English) Zbl 0897.06007
An element \(a\) of a \(C\)-lattice \(L\) is called
– \(P\)-weak meet principal if every prime element \(b\leq a\) is weak meet principal;
– \(P\)-principal if every prime element \(b\leq a\) is principal.
For a principally generated \(C\)-lattice, a characterization of the maximal, \(\Delta\)-prime, \(P\)-principal element is given and hence principally generated, principal element \(C\)-lattices are characterized. The properties of \(P\)-weak meet principal elements belonging to a \(C\)-lattice generated by weak join principal elements are derived. A sufficient condition on a lattice generated by weak join principal elements to be an almost principal element lattice is proven. Structure conditions for a lattice generated by compact weak join principal elements are found.

MSC:
06B05 Structure theory of lattices
06B15 Representation theory of lattices
06F10 Noether lattices
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