Domain理论中凸结构的研究

Domain Theory provides a mathematical foundation for the denotational semantics of functional programming languages in theoretical computer science. With the order relation as its core element, this theory can be considered as a mathematical branch involving structures of order, convexity, topology,logic, algebra, etc,. By using research skills in Convexity Theory and combining with the essential features of order structures in Domain Theory, the present project aims to study the properties of ordered convex sets and ordered convex structures induced by fuzzy order relation, and discuss the relationships among order-preserving mappings, convexity-preserving mappings and convex-to-convex mappings, and consequently try to characterize some domain structures and continuous fuzzy posets by means of their ordered convexity. We then generalize the approaches and results to algebraic structures with fuzzy orders and introduce the concept of fuzzy convex substructures with combining the definitions of fuzzy ordered convex sets and substructures. By analyzing the properties of fuzzy convex substructures, some convex structures in fuzzy ordered algebraic structures will be constructed, and some properties of convex spaces will be discussed. Finally, we will introduce a kind of generalized convex structure in hoping to unify the classical convex structures and the fuzzy convex structures in lattice-theoretic framework. A new order will be defined by using the generalized convex hull operator therein. The categorical isomorphism between the generalized convex spaces and the newly defined ordered spaces will be investigated.

Domain理论是计算机函数式程序语言指称语义的数学模型，它融合了序、凸、拓扑、逻辑和代数等多种结构，其上的序关系是Domain结构的灵魂和核心。本项目利用凸理论研究方法，结合Domain结构上的序关系本质特征，对模糊序关系诱导的凸集和凸结构的性质展开研究。讨论序结构意义下的保序映射和凸空间意义下的凸保持映射以及凸-凸映射之间的联系，并用序凸结构来等价刻画一些Domain结构和连续模糊偏序集。然后，我们把序凸结构的研究工作扩充到含有模糊序的代数结构上。结合模糊序凸集和子结构的定义引进模糊凸子结构的概念。通过分析模糊凸子结构的性质而构造出模糊序代数结构中的凸结构，并讨论凸空间的一些相关性质。最后，把经典凸结构和模糊凸结构统一到完备格框架下，引进广义凸结构。借助于广义凸包算子定义一种新的序关系，证明广义凸空间和新定义的序空间在范畴意义下同构。

本项目是凸理论、序关系和代数运算三个方面的融合研究。主要利用凸理论研究方法，结合序关系和代数运算自身的特点，在已有的序结构和序代数结构中来引进凸子集和凸子结构，进而研究相应的凸空间性质，并利用凸结构的性质反过来刻画序结构和序代数结构。研究内容主要体现在三个方面：首先在模糊偏序集上引进凸集和凸结构，给出模糊序凸结构的定义，并对相应的凸包算子和凸保持映射进行了详细的研究，得到了一些结论；从模糊二元运算出发重新定义了一种全新的序模糊群，研究其子群、正规子群、凸子群的性质和等价定理，并对这些子结构构造出来的相应凸空间性质进行了详尽的讨论；最后，我们把在集合基础上定义的经典凸结构和L-凸结构在完备格框架下有机地统一起来，引进了广义凸结构理论，反过来利用广义凸包算子定义了一种序关系，并且证明这种新的序空间和广义凸空间是范畴同构的。我们的研究内容同时丰富和发展了Domain理论和凸结构理论，把凸理论研究方法应用到序结构和序代数结构研究中，既促进了凸结构理论的进一步发展，同时为序结构的应用研究提供一种新思路、新方法，从而更好地为理论计算机科学的发展奠定数学基础。