The Lattice of Subalgebras of a Boolean Algebra
The Lattice of Subalgebras of a Boolean Algebra
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布尔代数的子代数格
DOI：
10.4153/cjm-1962-035-1
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发表时间：
1962
期刊：
影响因子：
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通讯作者：
David Sachs
中科院分区：
文献类型：
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作者：
David Sachs
众所周知（1，p.162），有限布尔代数的子代数格对偶同构于有限分格。本文研究了任意布尔代数的子代数格。我们的主要结果之一是子代数格刻画了布尔代数。为了证明这一结果，我们引入了一些概念，使我们能够给出一个表征和表示格的子代数的布尔代数的封闭算子的格的分区的布尔空间与布尔代数。我们的理论，然后有一些类似的拓扑向量空间的格理论。一些感兴趣的是问题的分类布尔代数的性质，他们的格的子代数，我们得到了一些结果在这个方向上。
It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.