【摘要】：Hoop代數(shù)最先由B.Bosbach于20世紀(jì)60年代作為一種自然序的可換剩余正半群而提出的.modal邏輯是非經(jīng)典邏輯范疇的一個重要分支,modal算子是直覺命題邏輯的代數(shù)語義.微分的思想源于分析學(xué).monadic算子是將謂詞邏輯中存在量詞和任意量詞進(jìn)行了代數(shù)化.本文將研究Hoop代數(shù)上的modal算子以及W-Hoop代數(shù)上的monadic微分算子.所做的工作如下:首先,我們在Hoop代數(shù)H上引入了 modal算子,并討論了相關(guān)性質(zhì).進(jìn)一步研究了H上的三個特殊映射,并給出這三個映射成為modal算子的等價刻畫.接下來,又從閉包算子的角度對modal算子進(jìn)行了深入研究,證明了兩個modal算子的復(fù)合是可換的的等價刻畫.此外,定義并研究了moda 可表示濾子,并得出在任何一個Hoop代數(shù)丑上,區(qū)間[a,1]是一個modal可表示濾子,其中a ∈ dmm(H).進(jìn)進(jìn)步,引入了modda同態(tài)與modal同余,并證明了兩個ModalHoop代數(shù)(H,f)與([0,a],fa)之間存在一個滿同態(tài),其中fa(x)=f(x)∧a,a ∈ Idm(H).最后,定義并研究了對偶modal算子,對偶modal濾子,并且證明了在Modal Hoop代數(shù)中,modal算子f和對偶modal算子f*之間可形成一個Galois聯(lián)結(jié).其次,我們在W-Hoop代數(shù)上將monadic算子和微分算子相結(jié)合進(jìn)行了研究,也就是在W-Hoop代數(shù)上研究了monadi 微分算子.具體來說就是在Monadic-Hoop代數(shù)(M,(?))上引入并研究了 M-微分.定義并研究了Monadic W-Hoop代數(shù)(M,(?))上的三類特殊微分——強(qiáng)M-微分,正則M-微分和可加M-微分,并利用這三類微分得到W-Hoop代數(shù)成為布爾代數(shù)的等價刻畫以及正則M-微分成為保序微分的等價刻畫.最后,在微分Monadic W-Hoop代數(shù)(M,(?),d)上定義了 monadic微分理想,并對其進(jìn)行了刻畫,而且研究了(M,(?),d)上所有monadic微分理想組成的集合LD(M)的代數(shù)結(jié)構(gòu),得到(ID(M),∧,∨,(?),M)是一個有界分配格.最后,我們在W-Hoop代數(shù)上研究了 moda算子和monadic微分算子之間的關(guān)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，

本文編號：2126967