论文标题
部分框架，免费框架和一致框架
Partial frames, their free frames and their congruence frames
论文作者
论文摘要
这项工作的背景是部分框架；这些是并非所有子集都需要加入的聚会 - 隔离。 SELECTION函数S，指定所有Meet-emilattices，即所考虑的某些子集，我们称为``指定''的子集；然后，S型框架必须加入（至少）所有此类子集和二进制聚会必须分配这些子集。一小部分公理足以指定我们的选择功能；这些公理足够通用，可以作为部分框架，有限的分布晶格，Sigma-Frames，Kappa-Frames和Frames的示例。我们将S框架地图的左右伴随视为引入封闭图和开放地图的前奏。 然后，我们查看部分框架可能是一个适当的布尔值概念。显而易见的候选人是每个元素得到补充的条件。这个概念确实引起了人们的关注，但是我们提出了三个进一步的条件，在框架设置中，这一切都等同于此。但是，在部分帧的背景下，四个条件是不同的。在调查这些过程中，我们在部分框架和部分框架的一致框架上对自由框架进行了必不可少的使用。 我们将部分框架（技术上称为s-congrence）的一致性与其自由框架的框架一致性进行了比较。我们为情况提供了自然的转换，还考虑了有关框架图的正确伴随。我们表征了两个一致性框架是同构并提供示例的情况，以阐明两者的不同行为。我们以封闭性和开放性的特征来结束，以将部分框架嵌入其自由名望中，并将其嵌入其一致性框架中。
The context of this work is that of partial frames; these are meet-semilattices where not all subsets need have joins. A selection function, S, specifies, for all meet-semilattices, certain subsets under consideration, which we call the ``designated'' ones; an S-frame then must have joins of (at least) all such subsets and binary meet must distribute over these. A small collection of axioms suffices to specify our selection functions; these axioms are sufficiently general to include as examples of partial frames, bounded distributive lattices, sigma-frames, kappa-frames and frames. We consider right and left adjoints of S-frame maps, as a prelude to the introduction of closed and open maps. Then we look at what might be an appropriate notion of Booleanness for partial frames. The obvious candidate is the condition that every element be complemented; this concept is indeed of interest, but we pose three further conditions which, in the frame setting, are all equivalent to it. However, in the context of partial frames, the four conditions are distinct. In investigating these, we make essential use of the free frame over a partial frame and the congruence frame of a partial frame. We compare congruences of a partial frame, technically called S-congruences, with the frame congruences of its free frame. We provide a natural transformation for the situation and also consider right adjoints of the frame maps in question. We characterize the case where the two congruence frames are isomorphic and provide examples which illuminate the possible different behaviour of the two. We conclude with a characterization of closedness and openness for the embedding of a partial frame into its free fame, and into its congruence frame.
