Order-Topological lattices
Order-Topological lattices
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DOI：
10.1017/s0017089500003980
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发表时间：
1980-01
影响因子：
0.5
通讯作者：
M. Erné
中科院分区：
文献类型：
--
作者：
M. Erné
观察到收敛的真实的序列可以定义在限制下级和限制上级作为通过邻里的欧几里德拓扑导致的问题：对于哪些格的顺序收敛符合收敛的顺序拓扑？这个问题已经被D. C.肯特，A. Gingras [7]和其他人。我们希望在本文中提出一个令人满意的解决方案。虽然有已知的几个特征的格，与拓扑序收敛（cf。命题1，2），这些标准的评估已经需要给定格的序拓扑的一些知识。在本文中，我们建立了一个纯粹的格理论的描述，这些格的顺序收敛不仅是拓扑的，而且，格操作是连续的。因此，这种格将被称为序拓扑格。所有的收敛性陈述都将用过滤器而不是网来表达。关于收敛函数的介绍，读者可以参考D. C. Kents的论文[9]。
The observation that convergence of real sequences may be defined in terms of limits inferior and limits superior as by means of neighbourhoods in the Euclidean topology leads to the question: for which lattices does order convergence coincide with convergence in the order topology? This problem has been attacked by D. C. Kent [10], A. Gingras [7] and others. We hope to present a satisfactory solution in this paper. Although there are known several characterizations of lattices, with topological order convergence (cf. Propositions 1, 2), an evaluation of these criteria already requires some knowledge of the order topology of the given lattice. In the present paper, we establish a purely lattice-theoretical description of those lattices for which order convergence is not only topological, but moreover, the lattice operations are continuous. Henceforth, such lattices will be referred to as order-topological lattices. All convergence statements will be formulated in terms of filters rather than nets. For an introduction to convergence functions, the reader may consult D. C. Kents's paper [9].