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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

$ n$-normal lattices


Author: William H. Cornish
Journal: Proc. Amer. Math. Soc. 45 (1974), 48-54
MSC: Primary 06A35
MathSciNet review: 0340133
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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$.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0340133-6
Keywords: Distributive lattice, minimal prime, $ n$-prime ideal, $ n$-normal lattice
Article copyright: © Copyright 1974 American Mathematical Society