S-lattices (Q2711242)

Let \(L\) be a multiplicative lattice, i.e., \(L\) is a not necessarily modular complete lattice with a commutative associative product that distributes over arbitrary joins and has compact greatest element \(I\) as a multiplicative identity. An element \(e\) of \(L\) is meet (join) principal if it satisfies \(a\wedge be= ((a: e)\wedge b)e\) \(((a\vee be): e=(a: e)\vee b)\) for all \(a,b\in L\). A principal element is an element that is both meet and join principal. A C-lattice is a multiplicative lattice which is generated under joins by a multiplicatively closed set of compact elements. Throughout this paper \(L\) denotes a C-lattice (join) generated by compact join principal elements. A prime element \(p\) of \(L\) is called an S-element if (1) the set of all \(p\)-primary elements is totally ordered, (2) \(p^\Delta\), the meet of all \(p\)-primary elements, is prime, and (3) \(p^\Delta\) contains each prime element properly contained in \(p\). Finally, \(F\) is an S-lattice if each prime element is an S-element. Several results concerning S-elements and S-lattices are given. For example, it is shown that if \(L\) is a principally generated reduced lattice, then for each maximal element \(m\) of \(L\), \(L_m\) has every element a principal element if and only if every prime of \(L\) is an S-element and is locally compact.