Generalized symmetric elements generated by a prime sequence (Q1966154)

Let \(A_1,A_2,\ldots ,A_n\) be a prime sequence in a local Noether lattice \(L\). Let, for \(k\geq 1\), \(\mathcal P_k b\) denote the set of finite joins in \(L\) of power products of the generalized symmetric elements of order \(k\) in \(A_1,A_2,\ldots ,A_n\) together with 0 and 1. In a previous paper [\textit{M. E. Detlefsen}, Pac. J. Math. 77, 365-379 (1978; Zbl 0423.06011)], the author has shown that for \(k=1\), \(\mathcal P_kb\) is a Noetherian distributive \(\pi \)-lattice domain. In the present paper it is proved that for \(n\leq 3\) and for any \(k\), \(\mathcal P_kb\) is a sub-\(\pi \)-domain of \(\mathcal P_1b\), and that for \(n\geq 4\) and \(k\geq 2\), \(\mathcal P_kb\) is not closed under the meet of \(\mathcal P_1b\).