EQ-代数上的内态算子及态BL-代数上的微分

发布时间：2018-05-15 05:09

本文选题：EQ-代数 + 内态算子　；参考：《西北大学》2017年硕士论文

【摘要】：EQ-代数作为高阶模糊逻辑的真值的代数结构,不仅为模糊型理论提供了更为广泛的真值代数结构,而且是剩余格的一般化.而各类子结构逻辑的代数语义均是以剩余格为基础建立的.本文研究了 EQ-代数上的内态算子及态BL-代数上的微分理论,为描述高阶模糊逻辑中命题的真值平均度提供了更一般的代数方法,主要研究内容如下:第三章研究了 EQ-代数上的内态算子.首先,建立了 EQ-代数上内态算子的公理体系(简记具有内态算子的EQ-代数为SEQ-代数).通过研究内态算子的性质,刻画了good EQ-代数.其次,探讨了SEQ-代数与态剩余格、态BCK-代数之间的联系,重点讨论了SEQ-代数(E,σ)与态EQ-代数(E,μ)的关系.随后,研究了EQ-代数上态与内态的关系.第四章研究了 SEQ-代数上的S-滤子理论.首先,引入SEQ-代数上S-滤子及S-前滤子概念,并讨论了特殊的SEQ-代数(E,μ)的S-滤子与相对应的EQ-代数σ(E)的滤子之间的联系.其次,探讨了特殊的次直不可约SEQ-代数.随后,研究了 SEQ-代数的所有S-前滤子所构成集合SPE(E,σ)的代数结构,得到当E为good EQ-代数或lEQ-代数时,SPE(E,σ)构成一个完备Brouwerian格.更进一步,对于lEQ-代数而言,当σ是忠实的和保→,SPE(E,σ)构成一个Heyting代数.最后,在SEQ-代数中引入集合A的σ-对偶零化子,用它给出态射good EQ-代数上极小素滤子的一个等价刻画.继而,用极小素滤子刻画了可表示的态射good EQ-代数.第五章研究了态BL-代数上的微分理论.在态BL-代数(A,σ)上引入(& ,V)-微分,研究了态BL-代数(A,σ)上(& ,V)-微分的性质.引入了态BL-代数(A,σ)上的正则微分和强微分,给出了正则强微分成为保序微分的等价条件.提出了态BL-代数上的主微分概念,讨论了全体主微分的代数结构,并利用Galois联结给出主微分的微分伴随.最后,讨论了态BL-代数(A,σ)上不动点集Adσ,得到若d为正则保序的强微分,则Aaσ为A的格理想.这些结果极大的丰富了态逻辑代数上的微分理论.
[Abstract]:As the algebraic structure of truth value of higher-order fuzzy logic, EQ-algebra not only provides a more extensive truth algebraic structure for fuzzy type theory, but also is a generalization of residual lattices. The algebraic semantics of all kinds of substructural logic are established on the basis of residual lattices. In this paper, we study the inner state operators on EQ- algebras and the differential theory on BL-algebras of states, which provide a more general algebraic method for describing the truth average degree of propositions in higher-order fuzzy logic. The main contents are as follows: in chapter 3, the inner state operators on EQ-algebras are studied. Firstly, an axiomatic system of inner state operators on EQ-algebras is established. By studying the properties of inner state operators, we characterize good EQ- algebras. Secondly, the relation between SEQ-algebras and state residue lattices, state BCK-algebras is discussed, and the relation between SEQ-algebras E, 蟽) and state EQ-algebras is discussed. Then, the relation between state and inner state on EQ-algebra is studied. In chapter 4, the S-filter theory on SEQ-algebra is studied. Firstly, the concepts of S- filter and S- prefilter on SEQ- algebra are introduced, and the relation between the S- filter of the special SEQ- algebra E, 渭) and the filter of the corresponding EQ- algebra 蟽-(E) is discussed. Secondly, we discuss the special subdirect irreducible SEQ-algebras. Then, the algebraic structure of the set SPEQE, 蟽) composed of all S- prefilters of SEQ- algebras is studied. When E is a good EQ- algebra or lEQ- algebra, a complete Brouwerian lattice is constructed. Furthermore, for lEQ- algebras, a Heyting algebra is constructed when 蟽 is faithful and preserved. Finally, the 蟽-dual annihilator of set A is introduced in SEQ- algebras, and an equivalent characterization of minimal prime filters on morphism good EQ-algebras is given. Then, we use the minimal prime filter to characterize the denoted morphism good EQ-algebras. In chapter 5, the differential theory on BL-algebra is studied. In this paper, we introduce a new type of differential on the state BL-algebra A, 蟽, and study the properties of the differential on the state BL-algebra A, 蟽. In this paper, we introduce the regular differential and strong differential on the state BL-algebra A, 蟽, and give the equivalent conditions for the regular strong differential to be order-preserving differential. In this paper, the concept of principal differential on state BL-algebra is proposed, and the algebraic structure of all principal differential is discussed, and the differential adjoint of principal differential is given by using Galois connection. Finally, the fixed point set Ad 蟽 on the BL-algebra A, 蟽) is discussed. It is obtained that a 蟽 is the lattice ideal of A if d is a strong differential with regular order-preserving. These results greatly enrich the differential theory in the logic algebra of states.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O141;O159

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