一种泛逻辑代数系统

文[3]给出了理想状态(广义相关系数h=0.5,广义自相关系数k=0.5)下泛逻辑的形式演绎系统B,证明了此系统是可靠的。该文提出理想状态下(h=k=0.5)泛逻辑学对应的代数系统-UB代数,给出它的一系列性质。证明了UB代数是一个交换剩余半群;进一步证明了U B代数与M V代数、正规FI代数是等价的。     

分配序列效应代数的理想和同余

分配的序列效应代数（简记为DSEA）,是指在一个效应代数上带有一种乘积运算并满足一定的条件。介绍了分配的序列效应代数中的左理想、右理想、理想、素理想和同余等概念,并且证明了满足（RDP）性质并且以1为乘积单位的分配序列效应代数是具有（RDP）性质的反格分配序列效应代数的子直积。     

效应代数水平和上的序列积

建立了一族可和序列效应代数的水平和上具有序列积的充分必要条件,进而给出了由序列效应代数的水平和构造序列效应代数的一种方法。     

格蕴涵代数中的极小素理想及α-理想

在格蕴涵代数中,证明了极小素理想与极小格素理想的等价性,继而给出了极小素理想与零化子的相互表示定理。提出了格蕴涵代数中的[α]-理想概念并给出其若干等价刻画,证明了极小素理想是[α]-理想。证明了全体素[α]-理想之集[Sα(L)]是一个紧的Stone空间,进一步给出[Sα(L)]分别是[T1、][T2]拓扑空间的充要条件。     

粗代数与三值Lukasiewicz代数

在粗糙集的代数方法研究中一个重要的方面是从粗糙集的偶序对(〈下近似集,上近似集〉)表示入手,通过定义偶序对的基本运算,从而构造出相应粗代数,并寻找能抽象表示偶序对性质的一般代数结构.其中最有影响的粗代数分别是粗双Stone代数、近似空间代数和粗Nelson代数,它们对应的一般代数分别是正则双Stone代数、预粗代数和半简单Nelson代数.文章证明了这三种粗代数都可以化为三值Lukasiewicz代数,从而将它们统一到了三值Lukasiewicz代数的框架下.并在此基础上,更直接地证明了一个近似空间中的所有粗糙集可构成一个三值Lukasiewicz代数.最后给出一个实例,说明了从一个信息系统得到其对应三值Lukasiewicz代数的过程.     

N-可分效应代数

引入了N-可分效应代数的定义,证明了N-可分效应代数是区间效应代数且N-可分效应代数可嵌入到可分效应代数中.     

字的组合的半群方法

该文利用半群方法给出了语言的一些代数性质.首先,讨论了稠密语言的半群结构,给出了包含语言w(wk)(其中w∈A+,k是正整数)的一个稠密语言类;证明了稠密正规语言包含一个字与一个稠密正规右酉幺半群的积.其次,讨论了自由幺半群的正规分支可分解性,证明了自由幺半群及正规分支可分解语言与正规分支可分解的后缀语言的积是正规分支可分解的;应用这些结果证明了Shyr和Yu关于正规语言的两个猜想.     

粗糙集概念与运算的布尔矩阵表示

建立了属性集与布尔矩阵以及逻辑方程组的解之间的关系;在此基础上给出了粗糙集理论中概念与运算的布尔矩阵表示;最后证明了属性约简在布尔矩阵和代数两种不同表示下是等价的。     

广义代数Riccati方程的一个比较定理

给出了一个关于广义代数Riccati方程的比较定理, 作为推论给出了广义代数Riccati方程最大解的存在性定理, 并且证明了该最大解也是强解, 改进了以往文献中的结论.     

有理解的最简图表示

对于“树代数”上方程组的求解问题是Prolg-Ⅱ的理论模式中基本问题之一.在很多场合下,我们需要考虑它的有理无穷解,为了能够在机器上处理这类解,引出了有关解的有穷表示问题.本文讨论了解的图表示.证明了存在有穷图表示的充要条件是该解为有理解;另外,本文还证明了,对于一个有理解,它的最简图表示总是存在的,并且在同构意义下是唯一的;最后本文给出了求最简图表示的有效算法.     

由可精确测量元控制的弱可换的伪效应代数

主要研究了由可精确测量元控制的弱可换的伪效应代数中可精确测量元。证明了可精确测量元控制的弱可换的伪效应代数中可精确测量元是弱可换的伪正交代数代数。讨论了弱可换的伪效应代数与BZ-偏序集之间的关系。讨论了弱可换的伪效应代数商代数中可精确测量元与正规Riesz理想之间的关系。     

Embeddings of generalized effect algebras into complete effect algebras

Generalized effect algebras as posets are unbounded versions of effect algebras having bounded effect-algebraic extensions. We show that when the MacNeille completion MC(P) of a generalized effect algebra P cannot be organized into a complete effect algebra by extending the operation ⊕ onto MC(P) then still P may be densely embedded into a complete effect algebra. Namely, we show these facts for Archimedean GMV-effect algebras and block-finite prelattice generalized effect algebras. Moreover, we show that extendable commutative BCK-algebras directed upwards are equivalent to generalized MV-effect algebras.     

Fuzzy maximal ideals of BCI and MV algebras

We consider fuzzy maximal ideals of BCI, BCK, and MV algebras, and show that such fuzzy ideals take only the values {0,1}, and have level ideals which are maximal ideals of the algebra. It is shown that fuzzy maximal ideals of commutative BCK algebras are fuzzy prime. We establish a correspondence between ideals and fuzzy ideals. In a bounded BCK algebra, this correspondence is one-to-one between maximal ideals and fuzzy maximal ideals. We also show how to construct fuzzy ideals which are not characteristic functions of ideals of an MV algebra.     

Riesz ideals in generalized pseudo effect algebras and in their unitizations

We prove that there is an order isomorphism between the lattice of all normal Riesz ideals and the lattice of all Riesz congruences in upwards directed generalized pseudoeffect algebras (or GPEAs, for short). We give a sufficient and necessary condition under which a normal Riesz ideal I of a weak commutative generalized pseudoeffect algebra P is a normal Riesz ideal also in the unitization [^(P)]\widehat{P} of P. These results extend those obtained recently by Avalllone, Vitolo, Pulmannová and Vinceková for effect algebras. At the same time, we give the conditions under which the quotient of a generalized pseudoeffect algebra P is a generalized effect algebra and linearly ordered generalized pseudoeffect algebra.     

Intervals in generalized effect algebras

A significant property of a generalized effect algebra is that its every interval with inherited partial sum is an effect algebra. We show that in some sense the converse is also true. More precisely, we prove that a set with zero element is a generalized effect algebra if and only if all its intervals are effect algebras. We investigate inheritance of some properties from intervals to generalized effect algebras, e.g., the Riesz decomposition property, compatibility of every pair of elements, dense embedding into a complete effect algebra, to be a sub-(generalized) effect algebra, to be lattice ordered and others. The response to the Open Problem from Rie?anová and Zajac (2013) for generalized effect algebras and their sub-generalized effect algebras is given.     

Computer simplification of formulas in linear systems theory

Currently, the three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA. These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Grobner basis algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations. On the other hand, most of the computation involved in linear control theory is performed on matrices, which do not commute, and Mathematica, Maple, and MACSYMA are weak in the area of noncommutative operations. The paper reports on applications of a powerful tool, a noncommutative version of the Grobner basis algorithm. The commutative version of this algorithm is implemented in most major computer algebra packages. The noncommutative version is relatively new     

On BCK algebras: Part II: New algebras. The ordinal sum (product) of two bounded BCK algebras

Since all the algebras connected to logic have, more or less explicitly, an associated order relation, it follows, by duality principle, that they have two presentations, dual to each other. We classify these dual presentations in “left” and “right” ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as “left” algebras or as “right” algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the “left” presentation and several algebras of logic have been redefined as particular cases of BCK algebras. We introduce several new properties of algebras of logic, besides those usually existing in the literature, which generate a more refined classification, depending on the properties satisfied. In this work (Parts I–V) we make an exhaustive study of these algebras—with two bounds and with one bound—and we present classes of finite examples, in bounded case. In Part II, we continue to present new properties, and consequently new algebras; among them, bounded α γ algebra is a common generalization of MTL algebra and divisible bounded residuated lattice (bounded commutative Rl-monoid). We introduce and study the ordinal sum (product) of two bounded BCK algebras. Dedicated to Grigore C. Moisil (1906–1973).     

双线性系统：对称代数构造

本文根据矩阵的Jordan标准形建立矩阵的交换代数的构造方法,在此基础上导出双线性系统的对称代数,并给出寻找一般非线性系统的对称群的一种途径。