Length of ideals in lattices

Autores: Ladislav Beran
Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 46, Fasc. 1-2, 1995 (Ejemplar dedicado a: Professor Paul Dubreil (In memoriam )), págs. 21-34
Idioma: inglés
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Resumen
- 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 nonnegative 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: {\footnotesize(i)} $l(I) = 1$ for any semiprime ideal $I$ of a lattice satisfying the ascending chain condition (briefly (ACC)); {\footnotesize(ii)} $l(I) = 1$ for any ideal $I$ of a distributive lattice satisfying the (ACC); {\footnotesize(iii)} $l(I)\leq 2$ for any ideal $I$ of a modular lattice having no infinite chains; and {\footnotesize(iv)} given any nonnegative integer $n$, there exists an ideal $I$ such that $l(I) = n$.