Hoop代数上两类逻辑算子的研究
发布时间：2018-07-16 16:32
【摘要】：Hoop代数最先由B.Bosbach于20世纪60年代作为一种自然序的可换剩余正半群而提出的.modal逻辑是非经典逻辑范畴的一个重要分支,modal算子是直觉命题逻辑的代数语义.微分的思想源于分析学.monadic算子是将谓词逻辑中存在量词和任意量词进行了代数化.本文将研究Hoop代数上的modal算子以及W-Hoop代数上的monadic微分算子.所做的工作如下:首先,我们在Hoop代数H上引入了 modal算子,并讨论了相关性质.进一步研究了H上的三个特殊映射,并给出这三个映射成为modal算子的等价刻画.接下来,又从闭包算子的角度对modal算子进行了深入研究,证明了两个modal算子的复合是可换的的等价刻画.此外,定义并研究了moda 可表示滤子,并得出在任何一个Hoop代数丑上,区间[a,1]是一个modal可表示滤子,其中a ∈ dmm(H).进进步,引入了modda同态与modal同余,并证明了两个ModalHoop代数(H,f)与([0,a],fa)之间存在一个满同态,其中fa(x)=f(x)∧a,a ∈ Idm(H).最后,定义并研究了对偶modal算子,对偶modal滤子,并且证明了在Modal Hoop代数中,modal算子f和对偶modal算子f*之间可形成一个Galois联结.其次,我们在W-Hoop代数上将monadic算子和微分算子相结合进行了研究,也就是在W-Hoop代数上研究了monadi 微分算子.具体来说就是在Monadic-Hoop代数(M,(?))上引入并研究了 M-微分.定义并研究了Monadic W-Hoop代数(M,(?))上的三类特殊微分——强M-微分,正则M-微分和可加M-微分,并利用这三类微分得到W-Hoop代数成为布尔代数的等价刻画以及正则M-微分成为保序微分的等价刻画.最后,在微分Monadic W-Hoop代数(M,(?),d)上定义了 monadic微分理想,并对其进行了刻画,而且研究了(M,(?),d)上所有monadic微分理想组成的集合LD(M)的代数结构,得到(ID(M),∧,∨,(?),M)是一个有界分配格.最后,我们在W-Hoop代数上研究了 moda算子和monadic微分算子之间的关系。
[Abstract]:Hoop algebra, first proposed by B. Bosbach as a commutative residual positive semigroup of natural order in the 1960s, is an important branch of nonclassical logic category and the algebraic semantics of intuitionistic propositional logic. The idea of differentiation originates from the algebraic transformation of the existential quantifiers and arbitrary quantifiers in predicate logic by the analytic .monadic operator. In this paper, we will study modal operators on Hoop algebras and monadic differential operators on W-Hoop algebras. The work is as follows: firstly, we introduce modal operator on the Hoop algebra H and discuss the related properties. In this paper, three special mappings on H are studied, and the equivalent characterizations of the three mappings as modal operators are given. Then, the modal operator is studied from the point of view of closure operator, and it is proved that the composition of two modal operators is commutative and equivalent. In addition, we define and study the moda representable filter, and obtain that the interval [a1] is a modal representable filter on any Hoop algebra, where a 鈭，
本文编号：2126967
[Abstract]:Hoop algebra, first proposed by B. Bosbach as a commutative residual positive semigroup of natural order in the 1960s, is an important branch of nonclassical logic category and the algebraic semantics of intuitionistic propositional logic. The idea of differentiation originates from the algebraic transformation of the existential quantifiers and arbitrary quantifiers in predicate logic by the analytic .monadic operator. In this paper, we will study modal operators on Hoop algebras and monadic differential operators on W-Hoop algebras. The work is as follows: firstly, we introduce modal operator on the Hoop algebra H and discuss the related properties. In this paper, three special mappings on H are studied, and the equivalent characterizations of the three mappings as modal operators are given. Then, the modal operator is studied from the point of view of closure operator, and it is proved that the composition of two modal operators is commutative and equivalent. In addition, we define and study the moda representable filter, and obtain that the interval [a1] is a modal representable filter on any Hoop algebra, where a 鈭，
本文编号：2126967
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