Beran, Ladislav Length of ideals in lattices. (English) Zbl 0842.06006 Collect. Math. 46, No. 1-2, 21-33 (1995). Summary: Let \(I^k\) be the \(k\)-th meander of an ideal \(I\) in a lattice \(L\) with 0 and 1. Define \(m\) to be the smallest nonegative integer such that \(I^m = I^{m+2}\) if such a number exists; in this case we put \(l(I) = m + 1\); otherwise we set \(l(I) = 0\). We show: (i) \(l(I) = 1\) for any semiprime ideal \(I\) of a lattice satisfying the ascending chain condition (briefly (ACC)); (ii) \(l(I) = 1\) for any ideal \(I\) of a distributive lattice satisfying the (ACC); (iii) \(l(I) = 2\) for any ideal \(I\) of a modular lattice having no infinite chains; and (iv) given any nonnegative integer \(n\), there exists an ideal \(I\) such that \(l(I) = n\). MSC: 06B10 Lattice ideals, congruence relations Keywords:meander of an ideal; semiprime ideal; ascending chain condition; distributive lattice; modular lattice × Cite Format Result Cite Review PDF Full Text: EuDML