Generalized continuous and hypercontinuous lattices
Generalized continuous and hypercontinuous lattices
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DOI：
10.1216/rmj-1981-11-2-271
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发表时间：
1981-06
影响因子：
0.8
通讯作者：
Hypercontinuous Lattices;J. Lawson
中科院分区：
文献类型：
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作者：
Hypercontinuous Lattices;J. Lawson
最近受到相当多关注的一类完全格是D. Scott[13]（参见[3]）引入的连续格。这类格的一个有趣的特征是，这些格承认一个唯一的紧致Hausdorff拓扑，其满足运算是连续的（即，它们承认紧致拓扑半格的结构）。这种拓扑是一种“内在”拓扑，即可以直接从晶格结构中定义的拓扑。我们将这种拓扑称为cl拓扑。本文的主要目标是对这种cl拓扑进行更详细的检查。对于任何完全格，这种拓扑结构总是紧的，并且是7\。我们对那些完备格进行表征，这些完备格是豪斯多夫的；由于这些格具有许多与连续格相似的特征，我们称它们为广义连续格。它们本身似乎是一类有趣的格；因此我们发展了它们的一些基本性质。Frink区间拓扑是最古老的内禀拓扑之一。我们要解决的问题是，对于哪些连续格，cl -拓扑和区间拓扑是重合的。这就是我们称之为“超连续”的格。我们将注意力转向这些格，并指出这些格与广义连续格之间的一些令人惊讶的联系。
A class of complete lattices which have recently received a considerable deal of attention is the class of continuous lattices introduced by D. Scott [13] (see also [3]). One of the interesting features of this class of lattices is the fact that these lattices admit a unique compact Hausdorff topology for which the meet operation is continuous (i.e., they admit the structure of a compact topological semilattice). This topology turns out to be an "intrinsic" topology, i.e., one that can be defined directly from the lattice structure. We refer to this topology as the CL-topology. A major goal of this paper is to give a more detailed examination of this CL-topology. For any complete lattice this topology is always compact and 7\. We characterize those complete lattices for which it is Hausdorff; because these lattices have many characteristics reminiscent of continuous lattices, we call them generalized continuous lattices. They seem to be an interesting class of lattices in their own right; hence we develop some of their fundamental properties. One of the oldest of the intrinsic topologies is Frink's interval topology. We address ourselves to the question of for what continuous lattices do the CL-topology and the interval topology coincide. This turns out to be precisely the class of lattices which we call "hypercontinuous". We turn our attention to these and point out some surprising connections between these lattices and generalized continuous lattices.