s-Prime elements in multiplicative lattices

Let be aC-lattice which is strong join principally generated. In this paper, we consider prime elements of 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 primepm has this property, then 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, then is the finite direct product of one-dimensional domains and primary lattices.