完备格上元素的分解及其在刻画无限Fuzzy关系方程解集中的应用

Fuzzy relational equations with sup-inf composition on complete Brouwerian lattices was first proposed by Sanchez in 1976, and was further studied by many researchers all over the world. However, how to describe the structure of solution sets of infinite fuzzy relational equations with sup-inf composition is still an open problem. The content of this project mainly includes: The infinite decomposition problems of elements on complete lattices; The condition that there exists minimal solutions、the construction methods of minimal solutions (resp. unattainable solutions) of infinite fuzzy relational equations with sup-inf compositon on complete Brouwerian lattices; The conditions that there exists partially attainable solutions, the construction methods of partially attainable solutions and the structure of solution sets; At last, we shall discuss the infinite fuzzy relational equations with non-commutative and non-associative composition and their structure of solution sets over complete Brouwerian lattices. The main idea of this project is to promote the further study of these problems and solve them.

Sanchez1976年提出完备Brouwer格上sup-inf合成Fuzzy关系方程的研究之后，国内外广大研究工作者做了大量的工作，但如何描述sup-inf合成无限Fuzzy关系方程解集的结构仍是公开问题.本项目的研究内容主要包括：完备格上元素的无限分解问题；完备Brouwer格上sup-inf合成无限Fuzzy关系方程极小解存在的条件以及极小解和不可达解的构造方法；sup-inf合成无限Fuzzy关系方程存在偏可达解的条件、偏可达解的构造方法及方程解集的结构；最后讨论完备Brouwer格上非交换、非结合合成算子的无限Fuzzy关系方程的求解问题,建立对应解集的结构.本项目旨在推动这些问题的深入研究和解决.

本项目主要讨论了完备格上元素的分解以及无限Fuzzy关系方程解集的结构问题。首先研究了完备Brouwer格上sup-conjunctor合成无限Fuzzy关系方程组，讨论可达解集,不可达解集与偏可达解集与方程的极小解之间的关系，存在可达解(极小解)的充要条件,极小解的构造方法以及方程组的可达解集的结构。进一步证明了不可达解集中的任何解都存在严格小于这个解的解，同时获得了sup-inf合成无限Fuzzy关系方程组的解集的性质和结构。对满足线性序和消去律的BL-代数上无限Fuzzy关系方程的解进行了研究，讨论了极大解与可达解、不可达解的关系; 对于不可达解部分，给出了方程的解结构特点; 对于可达解部分，找出了所有的极大解，证明了方程的每个可达解都能找到一个相应的极大解。对于完备格上元素的无限分解问题，获得了部分结果。项目最后还讨论了区间值模糊关系方程以及inf-implication合成模糊关系方程的极大解与覆盖问题的关系等问题。