Sub-effect algebras and Boolean sub-effect algebras

 We show that Boolean effect algebras may have proper sub-effect algebras and conversely. Properties of lattice effect algebras with two blocks are shown. One condition of the completness of effect algebras is given. We also show that a lattice effect algebra associated to an orthomodular lattice can be embedded into a complete effect algebra iff the orthomodular lattice can be embedded into a complete orthomodular lattice.     

Boolean Algebra as a Fragment of the Theory of Boolean Toposes

Basic identities of Boolean algebra are considered, and their categorical analogs are proved.     

Unification in Boolean rings

We show that two Boolean terms which are unifiable have a most general unifier, which can be described using the terms themselves and a single unifier. Techniques for finding a single unifier are given.     

Boolean dominated MV-algebras

In this paper we study an obvious generalization of the hyperarchimedean MV-algebras: boolean dominated MV-algebras. Particularly we point out the wide difference between the class of the hyperarchimedean MV-algebras and the class of the boolean dominated MV-algebras.     

A Shortest 2-Basis for Boolean Algebra in Terms of the Sheffer Stroke

In this article, we present a short 2-basis for Boolean algebra in terms of the Sheffer stroke and prove that no such 2-basis can be shorter. We also prove that the new 2-basis is unique (for its length) up to applications of commutativity. Our proof of the 2-basis was found by using the method of proof sketches and relied on the use of an automated reasoning program. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Complexity of Computation of Partial Derivatives of Boolean Functions Realized by Zhegalkin Polynomials

The combinational complexity of a system of partial derivatives in the basis of linear functions is established for a Boolean function of n variables that is realized by a Zhegalkin polynomial. An algorithm whose complexity equals 3ⁿ – 2ⁿ modulo 2 additions is proposed for computation of all partial derivatives of such a function from the coefficients of its Zhegalkin polynomial.     

Short Single Axioms for Boolean Algebra

We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.     

t-norms induced by metrics on Boolean algebras

Let d _ν be the metric associated with a strictly positive submeasure ν on some Boolean algebra . If d _ν is bounded from above by 1, E _ν=1−d _ν is a (fuzzy) similarity relation on at least w.r.t. the Lstrok ukasiewicz t-norm, but possibly also w.r.t. numerous further t-norms.In this paper, we show that under certain assumptions on and ν, we may associate with ν in a natural way a continuous t-norm w.r.t. which E _ν is a similarity relation and which, in a certain sense, is the weakest such t-norm. Up to isomorphism, every continuous t-norm arises in this way     

Boolean difference techniques in fault tree analysis

For an electrical, mechanical, or hybrid system described diagramatically as a network of interconnected components, fault tree modeling of system reliability as a function of individual component failure probabilities gives rise to logic expressions obtained from the network connections. Application of the method of Boolean differences in the analysis of such Boolean expressions is discussed, and it is shown that the influence of the status of specific components on the reliability of the total system may be investigated by straightforward algebraic operations on the network failure function.     

多值Boole过程

采用文献[1]中定义的扩展Allen-Givone代数概念将Boo1e过程沦扩充,提出了多值Boo1e过程的概念及其运算,为精确统一描述多值逻辑电路的逻辑功能和定时行为提供了一种解析途径。提出基于Allen-Givone代数的带状波形概念,用实值的加、减、乘、除运算为电路的异步特性提出了解析化的理论基础。这种数学分析与离散数学相结合的途径能相对精确地描述电路的时滞模型。在多值逻辑电路设计自动化技术的测试、模拟、综合等领域中,这种方法有它的应用前景。     

Rough 3-valued algebras

Rough set theory is an important tool for dealing with granularity and vagueness in information systems. This paper studies a kind of rough set algebra. The collection of all the rough sets of an approximation space can be made into a 3-valued Lukasiewicz algebra. We call the algebra a rough 3-valued Lukasiewicz algebra. In this paper, we focus on the rough 3-valued Lukasiewicz algebras, which are a special kind of 3-valued Lukasiewicz algebras. Firstly, we examine whether the rough 3-valued Lukasiewicz algebra is an axled 3-valued Lukasiewicz algebra. Secondly, we present the condition under which the rough 3-valued Lukasiewicz algebra is also a 3-valued Post algebra. Then we investigate the 3-valued Post subalgebra problem of the rough 3-valued Lukasiewicz algebra. Finally, this paper studies the relationship between the rough 3-valued Lukasiewicz algebra and the Boolean algebra constructed by all the exact sets of the corresponding approximation space.     

基于布尔语义的Gentzen推导模型

布尔模型是信息检索系统的一种基础模型。给出了命题逻辑和布尔代数间的一种新的对应关系,其中布尔代数中的不等式对应Gentzen系统中的矢列式,使得当一个不等式在任意布尔代数中为真,当且仅当它所对应的矢列式是可证的。并且使得在信息检索中,针对信息的推理可以有效地转为偏序集上的运算。讨论的命题逻辑语言的运算符为、ù、ú;并且定义了项(a|t|t1ùt2|t1út2其中a是一个元素)来替代原先的公式和表示布尔代数中的元素。此外,定义了以布尔代数为论域的赋值v,将命题逻辑中的项赋值为布尔代数中的元素,并且如果tΔΓt vtΔΔt v,则矢列式ΓT D为真。最后给出了Gentzen系统下的可靠性和完备性定理的证明。     

States on <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> algebras

The aim of this paper is to introduce the notion of states on R ₀ algebras and investigate some of their properties. We prove that every R ₀ algebra possesses at least one state. Moreover, we investigate states on weak R ₀ algebras and give some examples to show that, in contrast to R ₀ algebras, there exist weak R ₀ algebras which have no states. We also derive the condition under which finite linearly ordered weak R ₀ algebras have a state. This work is supported by NSFC (No.60605017).     

Parameter estimation for Boolean models of biological networks

Boolean networks have long been used as models of molecular networks, and they play an increasingly important role in systems biology. This paper describes a software package, Polynome, offered as a web service, that helps users construct Boolean network models based on experimental data and biological input. The key feature is a discrete analog of parameter estimation for continuous models. With only experimental data as input, the software can be used as a tool for reverse-engineering of Boolean network models from experimental time course data.     

Pseudo-BCK algebras and PD-posets

The notion of pseudo-BCK algebras was introduced by Georgescu and Iorgulescu Proceedings of DMTCS′01: Combinatorics, Computability and Logic, Springer, London, pp 97–114, 2001. It is so general that fleas, pseudo-hoops, psMTL, psBL, pseudo-MV et al., and so hoops, MTL, BL, MV et al. can be seen its extensions. In this paper, we extend the ideal and congruence theory to pseudo-BCK algebras, and investigate the connections between pseudo-BCK algebras and PD(GPD)-posets.     

Filter theory of BL algebras

In this paper we consider fundamental properties of some types of filters (Boolean, positive implicative, implicative and fantastic filters) of BL algebras defined in Haveshki et al. (Soft Comput 10:657–664, 2006) and Turunen (Arch Math Logic 40:467–473, 2001). It is proved in Haveshki et al. (2006) that if F is a maximal and (positive) implicative filter then it is a Boolean filter. In that paper there is an open problem Under what condition are Boolean filters positive implicative filters? One of our results gives an answer to the problem, that is, we need no more conditions. Moreover, we give simple characterizations of those filters by an identity form ? x, y(t(x, y) ∈ F), where t(x, y) is a term containing x, y.      

概率逻辑系统是与集合代数同态的布尔代数

联结词的本质是命题的运算,只有对所有命题都适用的真值函数才能用于定义联结词.概率逻辑中由于命题的内涵相关性,任何[0,1]上的函数都不能完全适用于任意命题的运算,概率逻辑的联结词不能定义成真值函数.各种算子可以作为一种计算方法使用和研究,但不能代表一个逻辑系统研究系统的性质.概率逻辑系统是概率空间的逻辑表示,是与概率空间中的事件域(集合代数)同态的布尔代数.用事件域上的集合函数精确定义各种联结词,与经典二值逻辑相容,与事实相符,能够在经典逻辑框架内实现概率命题演算.     

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).