﻿ 分配格的新定理 New Theorem for the Distributive Lattices

Pure Mathematics
Vol.05 No.02(2015), Article ID:14956,4 pages
10.12677/PM.2015.52012

New Theorem for the Distributive Lattices

Youxue Xu

Zhanjiang TV University, Zhanjiang Guangdong

Email: 529568449@qq.com

Received: Mar. 5th, 2015; accepted: Mar. 14th, 2015; published: Mar. 19th, 2015

ABSTRACT

This paper discusses the operational properties of generated ideals and generated dual ideals. Then an example is given to point out that the conclusion of a proposition on P10 reference [1] is wrong. Two equivalent conditions for the distributive lattices are given.

Keywords:Distributive Lattice, Generated Ideal, Generated Dual Ideal

Email: 529568449@qq.com

1. 生成幻及生成对偶幻的运算性质

，则

(1)是幻。

(2)

(3)是对偶幻。

(4)

(3) 设，有，使，由于闭于上知，，故

(4) 我们只要证：

(a) 设，则，不妨设，则对任意，有

(b) 显然。

(c) 设，则有，使

(d) 设，有，或

(1)

(2)

(2)是(1)的对偶。

(1)

(2)

(2)是(1)的对偶。

2. 分配格的两个等价条件

[1] 中P10指出：在格L中，由它的子集X，Y生成的幻的并(交)，易知是由所作全体生成的幻*。

(1)的含的最小幻。

(2)的含的最小幻。

Figure 1. Diamond lattice

Figure 2. Four elements of lattice

(3)

(4)

，则

New Theorem for the Distributive Lattices. 理论数学,02,80-84. doi: 10.12677/PM.2015.52012

1. 1. 中山 正 (1964) 格论. 董克诚译. 上海科学技术出版社, 上海.