$n$-normal lattices

Abstract:

An $n$-normal lattice is a distributive lattice with 0 such that each prime ideal contains at most $n$ minimal prime ideals. A relatively $n$-normal lattice is a distributive lattice such that each bounded closed interval is an $n$-normal lattice. The main results of this paper are: (1) a distributive lattice $L$ with 0 is $n$-normal if and only if for any ${x_0},{x_1}, \cdots ,{x_n}\varepsilon L$ such that ${x_i} \wedge {x_j} = 0$ for any $i \ne j,i,j = 0, \cdots ,n,{({x_0}]^ \ast } \vee {({x_1}]^ \ast } \vee \cdots \vee {({x_n}]^ \ast } = L$, (2) a distributive lattice $L$ is relatively $n$-normal if and only if for any $n + 1$ incomparable prime ideals ${P_0},{P_1}, \ldots ,{P_n},{P_0} \vee {P_1} \vee \ldots \vee {P_n} = L$.