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Length of ideals in lattices. (English) Zbl 0842.06006

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