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.     

Lattice pseudoeffect algebras as double residuated structures

Pseudoeffect algebras are partial algebraic structures which are non-commutative generalizations of effect algebras. The main result of the paper is a characterization of lattice pseudoeffect algebras in terms of so-called pseudo Sasaki algebras. In contrast to pseudoeffect algebras, pseudo Sasaki algebras are total algebras. They are obtained as a generalization of Sasaki algebras, which in turn characterize lattice effect algebras. Moreover, it is shown that lattice pseudoeffect algebras are a special case of double CI-posets, which are algebraic structures with two pairs of residuated operations, and which can be considered as generalizations of residuated posets. For instance, a lattice ordered pseudoeffect algebra, regarded as a double CI-poset, becomes a residuated poset if and only if it is a pseudo MV-algebra. It is also shown that an arbitrary pseudoeffect algebra can be described as a special case of conditional double CI-poset, in which case the two pairs of residuated operations are only partially defined.     

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.     

乘积格蕴涵代数的子代数的MATLAB实现

借助于MATLAB,得到了两个有限链型的乘积格蕴涵代数子代数的具体形式,更直观地反映了格蕴涵子代数的结构特征,并且通过一个例子展示了程序的运行结果。研究了格蕴涵代数中各种滤子之间的关系,讨论了两个链型格蕴涵子代数乘积的滤子,得到了其只存在平凡滤子和素滤子的结论。     

Lattice implication ordered semigroups

From the viewpoint of semantics, lattice implication algebras provide a basis to establish lattice-valued logic with truth value in a relatively general lattice. In this paper, we first introduce two notions of lattice implication n-ordered semigroup and lattice implication p-ordered semigroup, which induced by lattice implication algebras. Secondly, we study some of their basic properties and prove that a lattice implication n-ordered semigroup is a residuated semigroup, and a lattice implication p-ordered semigroup is an arithmetic lattice ordered semigroup. We also define the homomorphism mapping between lattice implication n-ordered semigroups. Finally, we discuss some properties of filters and sl ideals in lattice implication n-ordered semigroups and lattice implication p-ordered semigroups.     

Lattice structure on some fuzzy algebraic systems

In this paper, we study the lattice structure of some fuzzy algebraic systems such as (G-)fuzzy groups, some fuzzy ordered algebras and fuzzy hyperstructures. We prove that under suitable conditions, these structures form a distributive or modular lattice. This research partially is supported by the “ Fuzzy Systems and its Applications Center of Excelence, Shahid Bahonar University of Kerman, Iran”.     

Ideals and quotients in lattice ordered effect algebras

 We show that a quotient of a lattice ordered effect algebra L with respect to a Riesz ideal I is linearly ordered if and only if I is a prime ideal, and the quotient is an MV-algebra if and only if I is an intersection of prime ideals. A generalization of the commutators in OMLs is defined in the frame of lattice ordered effect algebras, such that the quotient with respect to a Riesz ideal I is an MV-algebra if and only if I contains all generalized commutators. If L is an OML, generalized commutators coincide with the usual Marsden commutators.     

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.     

Unique fixed points vs. least fixed points

The aim of this paper is to compare two approaches to the semantics of programming languages: the least fixed point approach, and the unique fixed point approach. Briefly speaking, we investigate here the problem of existence of extensions of algebras with the unique fixed point property to ordered algebras with the least fixed point property, that preserve the fixed point solutions. We prove that such extensions always exist, the construction of a free extension is given. It is also shown that in some cases there is no ‘faithful’ extension, i.e. some elements of a carrier are always collapsed.     

From quantitative to qualitative orness for lattice OWA operators

This paper deals with OWA (ordered weighted average) operators defined on an arbitrary finite lattice endowed with a t-norm and a t-conorm. A qualitative orness measure for any OWA operator is suggested, based on its proximity to the OR operator that yields the maximum of the given data. In the particular case of a finite distributive lattice, considering the t-norm given by the meet and the t-conorm given by the join, this qualitative measure agrees with the value that some discrete Sugeno integral takes on the vector consisting of all the members of the lattice. Some applications of the qualitative orness of OWA operators to decision-making problems are shown. In addition, OWA operators defined on a finite product lattice are also applied in image processing. We analyze the effect of several OWA operators with respect to their orness.     

赋范格H蕴涵代数和模糊格H蕴涵代数

在范数的条件下扩充了格H蕴涵代数的概念,即赋范格H蕴涵代数,并讨论一些性质。然后将模糊集合论运用于赋范格H蕴涵代数,给出了模糊赋范格H蕴涵代数的定义,得到了一些基本性质。通过使用两个赋范格H蕴涵代数之间的映射定义了赋范格H蕴涵代数同态,且得到了一些性质。最后得到了在赋范格H蕴涵代数中的数列对于蕴涵距离是有界的结论。     

Partial residuated structures and quantum structures

Residuated structures, bounded commutative residuated lattices in particular, play an important role in the study of algebraic structures of logics—classical and non-classical. In this paper, by introducing partial adjoint pairs, a new structure is presented, named partial residuated lattices, which can be regarded as a version of residuated lattices in the case of partial operations, and their basic properties are investigated. The relations between partial residuated lattices and certain quantum structures are considered. We show that lattice effect algebras and D-lattices both are partial residuated lattices. Conversely, under certain conditions partial residuated lattices are both lattice effect algebras and D-lattices. Finally, dropping the assumption on commutativity, some similar results are obtained. Project supported by the NSF of China (No. 10771524).     

On the algebraic specification of infinite objects — ordered and continuous models of algebraic types

Summary The concept of algebraic types is adapted to allow axiomatic characterizations of ordered and continuous algebras; infinite objects are then limit points in the carriers of certain continuous algebras. We mainly study implicative types, i.e., types the axioms of which are conditional inequations describing partial orders. The isomorphism classes of termgenerated ordered models of an implicative type are shown to form a complete lattice under the homomorphism ordering; this includes the wellknown initiality results for equational and conditional types as special cases. For types whose axioms specify strictness of the operations, the initial models are shown to correspond to flat domains.As a special kind of continuous algebras we consider inductively generated algebras, viz. the ideal completions of term-generated ordered algebras. For an inequational type, i.e., an implicative type where all axiom premises are empty, the completion of models always yields models again, whereas for implicative types this holds only in restricted cases. One such case is provided by hierarchic types which are complete and consistent relative to their primitive parts, and which satisfy certain conditions about limit points.Examples of algebras that can be specified by such types include those of finite and infinite streams, of sets of atoms under the Egli-Milner ordering, Milner's synchronisation trees, and that of a simple functional language over the natural numbers.This research was carried out within the Sonderforschungsbereich 49 Programmiertechnik, Munich, Federal Republic of Germany     

On lattice ordered soft sets

Certain type of linguistic terms such as satisfactory, good, very good and excellent have an order among them. In this paper we introduce a new concept of soft sets with some order among the parameters. Some properties of lattice ordered soft sets are given. Lattice ordered soft sets are very useful in particular type of decision making problems where some order exists among the elements of parameters set.     

序决策形式背景的规则提取及属性约简*

基于含有多值决策信息的决策形式背景,提出序决策形式背景的概念及其序决策概念格的相关理论,给出序决策概念格的决策规则及规则的置信度与支持度,并讨论决策规则在实际应用中的意义.在此基础上定义保持规则不变的属性约简,同时得到保持序决策概念格结构不变的属性约简方法.最后讨论序决策形式背景保持规则不变的约简与保持格结构不变的约简之间的关系.     

基于Unsharp量子逻辑的自动机和文法理论

初步建立了具有某种分配律的扩展格序效应代数和格序QMV代数这两种unsharp量子结构上的自动机与文法理论的基本框架。引入了ε-值正则文法的概念,证明了任意ε-值自动机识别的语言等价于某种ε-值正则文法所生成的语言;反之,任意[ε]-值正则文法所生成的语言等价于某种ε-值自动机识别的语言。讨论了ε-值正则语言在和、连接及反转运算下的封闭性质。     

Set space diagrams

This paper introduces set space diagrams and defines their formal syntax and semantics. Conventional region based diagrams, like Euler circles and Venn diagrams, represent sets and their intersections by means of overlapping regions. By contrast, set space diagrams provide a certain layout that avoids overlapping geometrical entities. This enables the representation of a good deal of sets without getting diagrams which are cluttered due to overlapping regions. In particular, these diagrams can be employed for illustration purposes, e.g., for showing the laws of Boolean algebras. Additionally, cardinalities are represented and can be easily compared; inferences can be drawn to derive unknown cardinalities from a given knowledge base. The soundness of set space diagrams is shown with respect to their set-theoretic interpretation.     

序列效应代数的Holland理论

介绍了序列效应代数的概念,给出了交换的序列效应代数的一些性质,证明了交换的序列效应代数由素理想诱导的商代数仍然是交换的序列效应代数。最后证明了每一个交换的序列效应代数能被表示为某个反格的自态射形成的序列效应代数。这样的表示是有用的,因为它给出了一个交换的序列效应代数作为自态射的集合的具体化。     

Path algebras: A useful CAI/CAL analysis technique

Networks in various forms are used extensively in Computer Aided Instruction and learning systems. Their use extends from simple frame instruction sequences to semantic networks and transition diagrams.The Path Algebra approach is a powerful mathematical tool for analysing networks. Different algebras may be defined to solve different problems. They have been successfully used in analysing man-machine dialogues and it is suggested that they may provide a useful analytical tool for CAI/CAL designers. Examples are given of algebras which are useful for analysing connectivity, step length, minimum paths, simple and elementary paths and for determining cut sets of arcs.     

Semirings and pseudo MV algebras

In this paper, we describe the relationships between pseudo MV algebras and semirings. We also give definitions of automata on lattice ordered semirings, prove that the family of K-Languages is closed under union, and discuss the conditions for the closedness of families of K-languages under intersection, generalized intersection and reversal operations.