Contact Algebras and Region-based Theory of Space: A Proximity Approach - I
Contact Algebras and Region-based Theory of Space: A Proximity Approach - I
复制标题
联系代数和基于区域的空间理论：邻近方法 - I
DOI：
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发表时间：
2006
影响因子：
0.8
通讯作者：
D. Vakarelov
中科院分区：
文献类型：
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作者：
G. Dimov;D. Vakarelov
本文是论文[2]的第二部分。它们都属于基于区域的空间理论（或称Whitehedian空间理论）领域，是定性空间推理（QSR）的一个重要分支。本文也可以被认为是抽象代数和拓扑学的一些问题的理论计算机科学和QSR中产生和动机的应用。在文献[2]中，给出了基于区域的空间理论的不同公理化。最一般的一个是在“接触代数”的名称下介绍的。本文引入了用切触代数语言定义的几个范畴。证明了它们等价于所有半正则T$_0$-空间及其连续映射的范畴，等价于以所有正则（分别为完全正则、紧、局部紧）Hausdorff空间为对象的满子范畴.本文给出了一个直接构造n阶半正则T$0 $-空间直到同胚的算法，并给出了一个不具有正则Hausdorff表示空间的RCC模型的例子。在这两个部分的主要调查方法是从邻近空间理论的方法和结构格理论的推广。给出了各种接触代数的邻近模型。这样，本文可以看作是对基于区域的空间理论的邻近方法的一个充分实现。
This paper is the second part of the paper [2]. Both of themare in the field of region-based (or Whitehedian) theory of space, which is an important subfield of Qualitative Spatial Reasoning (QSR). The paper can be considered also as an application of abstract algebra and topology to some problems arising and motivated in Theoretical Computer Science and QSR. In [2], different axiomatizations for region-based theory of space were given. The most general one was introduced under the name "Contact Algebra". In this paper some categories defined in the language of contact algebras are introduced. It is shown that they are equivalent to the category of all semiregular T$_0$-spaces and their continuous maps and to its full subcategories having as objects all regular (respectively, completely regular; compact; locally compact) Hausdorff spaces. An algorithm for a direct construction of all, up to homeomorphism, finite semiregular T$_0$-spaces of rank n is found. An example of an RCC model which has no regular Hausdorff representation space is presented. The main method of investigation in both parts is a lattice-theoretic generalization of methods and constructions from the theory of proximity spaces. Proximity models for various kinds of contact algebras are given here. In this way, the paper can be regarded as a full realization of the proximity approach to the region-based theory of space.