软BCK代数

提出了软BCK代数的概念,研究了软BCK代数的并,交和乘积运算,讨论了这些运算的代数性质。     

软BCK代数再研究

给出了软BCK代数的广义交运算和广义并运算,证明了一个集合X上的一族软BCK代数{(F_i,A_i)}_#em/em#∈I的广义交和一族软BCK代数{(F_i,A_i)}_#em/em#∈I（对任意#em/em#,j∈I,#em/em#≠j有A_i∩A_j=?)的广义并仍然是X上的软BCK代数,集合X上的软BCK代数(G,B)的一族软子代数的交、广义交和广义并仍是(G,B)的软子代数。     

粗糙集与BCK代数

粗糙集的代数刻画是粗糙集理论中研究的一个重要方向,在偶序对[R(X),R(X)]构成的集合中定义了二元运算[*],证明了序偶对构成的集合构成了一个有界BCK代数,给出了基本的性质,为粗糙逻辑和粗糙推理奠定了基础。     

On two versions of the Loomis–Sikorski Theorem for algebraic structures

We present two versions of the Loomis–Sikorski Theorem, one for monotone σ-complete generalized pseudo effect algebras with strong unit satisfying a kind of the Riesz decomposition property. The second one is for Dedekind σ-complete positive pseudo Vitali spaces with strong unit. For any case we can find an appropriate system of nonnegative bounded functions forming an algebra of the given type with the operations defined by points that maps epimorphically onto the algebra. The paper has been supported by the Center of Excellence SAS—Physics of Information—I/2/2005, the grant VEGA No. 2/6088/26 SAV, the Slovak Research and Development Agency under the contract No. APVV-0071-06, Slovak-Italian Project No. 15:“Algebraic and logical systems of soft computing”, and MURST, project “Analisi Reale”.     

Applications of soft sets in ideal theory of BCK/BCI-algebras

The notion of soft ideals and idealistic soft BCK/BCI-algebras is introduced, and several examples are given. Relations between soft BCK/BCI-algebras and idealistic soft BCK/BCI-algebras are provided. The intersection, union, “AND” operation, and “OR” operation of soft ideals and idealistic soft BCK/BCI-algebras are established.     

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.     

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.     

BCK/BCI-代数的广义交软理想

为了进一步研究交软BCK/BCI-理想的性质,通过在交软BCK/BCI-理想的概念中引入一个参数[α],从而给出了[α]-交软BCK/BCI-理想的定义,并在一定程度上推广了交软BCK/BCI-理想的概念。给出了[α]-交软BCK/BCI-理想的刻画方法和性质。讨论了[α]-交软BCK/BCI-理想在同态下像和逆像性质。     

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.     

Iséki algebras. Connection with BL algebras

We introduce Iséki algebras as a special class of BCK algebras and we prove they are categorically equivalent with BL algebras. Other connections are established.I thank very much Professor Antonio Di Nola for his kind suggestions that helped me to improve the final form of the paper.     

On Action Logic: Equational Theories of Action Algebras

Pratt (1991, Proceedings of JELIA’90, Volume 478, pp.97–120) defines action algebras as Kleene algebras withresiduals and action logic as the equational theory of actionalgebras. In contrast to Kleene algebras, action algebras forma (finitely based) variety. Jipsen (2004, Studia Logica, 76,291–303) proposes a Gentzen-style sequent system for actionlogic but leaves it as an open question if this system admitscut-elimination and if action logic is decidable. We show thatJipsen's system does not admit cut-elimination. We prove thatthe equational theory of *-continuous action algebras and thesimple Horn theory of *-continuous Kleene algebras are not recursivelyenumerable and they possess FMP, but action logic does not possessFMP.     

Generalizations of -fuzzy subalgebras in BCK/BCI-algebras

More general form of the notion of quasi-coincidence of a fuzzy point with a fuzzy set is considered, and generalizations of results in the papers [Y. B. Jun, On (α,β)-fuzzy subalgebras of BCK/BCI-algebras, Bull. Korean Math. Soc. 42 (4) (2005) 703–711; Y. B. Jun, Fuzzy subalgebras of type (α,β) in BCK/BCI-algebras, Kyungpook Math. J. 47 (2007) 403–410] are discussed. The notions of -fuzzy subalgebras and -fuzzy subalgebras in a BCK/BCI-algebra X are introduced, and several properties are investigated. Characterizations of -fuzzy subalgebra in a BCK/BCI-algebra X are discussed.     

Isomorphism theorems on generalized effect algebras based on atoms

A well-known fact is that every generalized effect algebra can be uniquely extended to an effect algebra in which it becomes a sub-generalized effect algebra and simultaneously a proper order ideal, the set-theoretic complement of which is its dual poset. We show that two non-isomorphic generalized effect algebras (even finite ones) may have isomorphic effect algebraic extensions. For Archimedean atomic lattice effect algebras we prove “Isomorphism theorem based on atoms”. As an application we obtain necessary and sufficient conditions for isomorphism of two prelattice Archimedean atomic generalized effect algebras with common (or isomorphic) effect algebraic extensions.     

Blocks of pseudo-effect algebras with the Riesz interpolation property

 Pseudo-effect algebras are partial algebras (E;+,0,1) with a partially defined sum + which is not necessary commutative only associative and with two complements, left and right ones. They are a generalization of effect algebras and of orthomodular posets as well as of (pseudo) MV-algebras. We define three kinds of compatibilities of elements and we show that if a pseudo-effect algebra satisfies the Riesz interpolation property, and another natural condition, then every maximal set of strongly compatible elements, called a block, is a pseudo MV-subalgebra, and the pseudo-effect algebra can be covered by blocks. Blocks correspond to Boolean subalgebras of orthomodular posets. Dedicated to Prof. Ján Jakubík on the occasion of his 80th birthday The paper has been supported by the grant VEGA 2/3163/23 SAV, Bratislava, Slovakia, and the fellowship of the Alexander von Humboldt Foundation, Bonn, Germany. The author is thankful the Alexander von Humboldt Foundation for organizing his stay at University of Ulm, Ulm, summer 2001, and Prof. G. Kalmbach H.E. for her cordial hospitality and discussions.     

Lower bounds for the bilinear complexity of associative algebras

Let R(A) denote the bilinear complexity (also called rank) of a finite dimensional associative algebra A.?We prove that if the decomposition of into simple algebras contains only noncommutative factors, that is, the division algebra is noncommutative or . In particular, -matrix multiplication requires at least essential bilinear multiplications. We also derive lower bounds of the form essential bilinear multiplications. We also derive lower bounds of the form for the algebra of upper triangular -matrices and the algebra of truncated bivariate polynomials in the indeterminates X,Y over some field k.?A class of algebras that has received wide attention in this context con-sists of those algebras A for which the Alder—Strassen Bound is sharp, i.e., R(A) = 2dim A—t is the number of maximal twosided ideals in A. These algebras are called algebras of minimal rank. We determine all semisimple algebras of minimal rank over arbitrary fields and all algebras of minimal rank over algebraically closed fields. Received: January 12, 2000.     

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).     

Region–based theory of discrete spaces: A proximity approach

We introduce Boolean proximity algebras as a generalization of Efremovič proximities which are suitable in reasoning about discrete regions. Following Stone’s representation theorem for Boolean algebras, it is shown that each such algebra is isomorphic to a substructure of a complete and atomic Boolean proximity algebra. Co-operation was supported by EC COST Action 274 “Theory and Applications of Relational Structures as Knowledge Instruments” (TARSKI), , and NATO Collaborative Linkage Grant PST.CLG 977641.     

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.     

Structured large margin machines: sensitive to data distributions

This paper proposes a new large margin classifier—the structured large margin machine (SLMM)—that is sensitive to the structure of the data distribution. The SLMM approach incorporates the merits of “structured” learning models, such as radial basis function networks and Gaussian mixture models, with the advantages of “unstructured” large margin learning schemes, such as support vector machines and maxi-min margin machines. We derive the SLMM model from the concepts of “structured degree” and “homospace”, based on an analysis of existing structured and unstructured learning models. Then, by using Ward’s agglomerative hierarchical clustering on input data (or data mappings in the kernel space) to extract the underlying data structure, we formulate SLMM training as a sequential second order cone programming. Many promising features of the SLMM approach are illustrated, including its accuracy, scalability, extensibility, and noise tolerance. We also demonstrate the theoretical importance of the SLMM model by showing that it generalizes existing approaches, such as SVMs and M⁴s, provides novel insight into learning models, and lays a foundation for conceiving other “structured” classifiers. Editor: Dale Schuurmans. This work was supported by the Hong Kong Research Grant Council under Grants G-T891 and B-Q519.     

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.