Generalized symmetric elements generated by a prime sequence

Abstract

Let \(A_1, A_2, \dots, A_n\) be a prime sequence in a local Noether lattice L. For \(k \ge 1, {\cal P} {_k}b \) denotes the set of finite joins in L of power products of the generalized symmetric elements of order k (majorization elements) in \(A_1, A_2, \dots, A_n\) together with 0 and I. We have previously showed that for \(k = 1, {\cal P} {_k}b \) is a Noetherian distributive \(\pi\)-domain. For \(n \le 3\) and for any \(k,{\cal P} {_k}b\) is again such a sub-\(\pi\)-domain of \({\cal P} {_1}b\). For \(n \ge 4\) and \(k \ge 2, {\cal P} {_k}b\) is not closed under the meet of \( {\cal P} {_1}b\). However \( {\cal P} {_k}b\) with its induced meet is again a Noetherian distributive \(\pi\)-domain. Each finite set of majorization elements asymptotically forms a distributive sublattice of \({\cal P}{_k}b\) for k sufficiently large.