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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
##### Keywords:
multiplicative lattice; principal element
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##### References:
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