Abstract
Let\(\mathcal{L}\) be aC-lattice which is strong join principally generated. In this paper, we consider prime elements of\(\mathcal{L}\) for which every semiprimary element is primary. We show, for example, that a compact nonmaximal primep with this property is principal. We also show that if every primep≤m has this property, then\(\mathcal{L}_m \) is either a one dimensional domain or a primary lattice. It follows that if every primep satisfies the property, and if there are only a finite number of minimal primes in\(\mathcal{L}\), then\(\mathcal{L}\) is the finite direct product of one-dimensional domains and primary lattices.
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Jayaram, C., Johnson, E.W. s-Prime elements in multiplicative lattices. Period Math Hung 31, 201–208 (1995). https://doi.org/10.1007/BF01882195
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DOI: https://doi.org/10.1007/BF01882195