Tight Galois connections and complete distributivity
Tight Galois connections and complete distributivity
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紧伽罗瓦连接和完全分布性
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发表时间：
1960
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通讯作者：
G. N. Raney
中科院分区：
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作者：
G. N. Raney
并讨论了它与完全格中完全分配性性质的关系。提出了在完整格子之间构建伽罗瓦连接的过程。通过此过程构造的伽罗瓦连接称为紧伽罗瓦连接，其特征是满足某些恒等式。完整格子上的所有闭包操作都可以通过紧伽罗瓦连接获得。如果伽罗瓦连接中涉及的完整格中的任何一个是完全分布的，则伽罗瓦连接是紧的。因此，由已知的 Birkhoff 过程构造的所有伽罗瓦连接都是紧的。从完全格到其对偶格的恒等映射总是确定伽罗瓦连接；当且仅当完整格是完全分布的时，这种伽罗瓦联系才是紧的。最后的观察结果仅根据其偏序来表征完全分布的完全格。它还提供了对这些格子结构的新见解，并使我们能够证明一种表示定理，该表示定理比以前已知的定理要经济得多。 2. 定义和符号。如果F是集合的子集族，则F的交集用HF表示，F的并集用EF表示。如果 L 是完全格，则 L 的每个子集 K 都有一个交集（用 nK 表示）和一个连接（用 UK 表示）。
and discusses its relations with the property of complete distributivity in complete lattices. A procedure for constructing Galois connections between complete lattices is presented. The Galois connections constructed by this procedure are called tight Galois connections, and are characterized as those which satisfy certain identities. All closure operations on complete lattices are obtainable from tight Galois connections. If either of the complete lattices involved in a Galois connection is completely distributive, then the Galois connection is tight. Consequently, all Galois connections constructed by the known procedure of Birkhoff are tight. The identity mapping from a complete lattice to its dual lattice always determines a Galois connection; this Galois connection is tight if and only if the complete lattice is completely distributive. This last observation leads to a characterization of completely distributive complete lattices solely in terms of the partial ordering on them. It also provides new insight into the structure of these lattices, and enables us to prove a representation theorem which is considerably more economical than the one previously known. 2. Definitions and notations. If F is a family of subsets of a set, the intersection of F is denoted by HF, and the union of F is denoted by EF. If L is a complete lattice, then every subset K of L has a meet, which is denoted by nK, and a join, which is denoted by UK.