二元Boolean矩阵的加权Moore-Penrose逆

主要研究了二元Boolean矩阵A的加权Moore-Penrose逆的存在性问题,给出了二元Boolean矩阵A的加权Moore-Penrose逆存在的一些充分必要条件,并讨论了加权Moore-Penrose逆存在时的若干等价刻画及惟一性问题.     

Generalized Boolean Algebras in Lattice-Ordered Groups

In this paper, characterizations are given for the free lattice-ordered group over a generalized Boolean algebra and the freel -module of a totally ordered integral domain with unit over a generalized Boolean algebra. Extensions of lattice-ordered groups using generalized Boolean algebras are defined and their properties studied.     

Inverses of Boolean matrices

For a Boolean matrix A, a g-inverse of A is a Boolean matrix G satisfying AGA=A, and a Vagner inverse is a g-inverse which in addition satisfies GAG=G. We give algorithms for finding all g-inverses, all Vagner inverses, and all of several other types of inverses including Moore-Penrose inverses. We give a criterion for a Boolean matrix to be regular, and criteria for the various types of inverse to exist. We count the numbers of Boolean matrices having Moore-Penrose and related types of inverses.     

On the semi-continuity of generalized inverses in Banach algebras

The main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a Banach algebra with identity. We first obtain a sufficient and necessary condition for the lower semi-continuity of reflexive generalized inverses as set-valued mappings. Based on this result, we characterize the continuity of the Moore-Penrose inverse in a C^∗-algebra and therefore, derive some new and well-known criteria in operator theory.     

Algebraic aspects of generalized approximation spaces

The concept of approximation spaces is a key notion of rough set theory, which is an important tool for approximate reasoning about data. This paper concerns algebraic aspects of generalized approximation spaces. Concepts of R-open sets, R-closed sets and regular sets of a generalized approximation space (U,R) are introduced. Algebraic structures of various families of subsets of (U,R) under the set-inclusion order are investigated. Main results are: (1) The family of all R-open sets (respectively, R-closed sets, R-clopen sets) is both a completely distributive lattice and an algebraic lattice, and in addition a complete Boolean algebra if relation R is symmetric. (2) The family of definable sets is both an algebraic completely distributive lattice and a complete Boolean algebra if relation R is serial. (3) The collection of upper (respectively, lower) approximation sets is a completely distributive lattice if and only if the involved relation is regular. (4) The family of regular sets is a complete Boolean algebra if the involved relation is serial and transitive.     

Implicative twist-structures

The twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety.     

关于布尔矩阵的广义Moore-Penrose逆

讨论布尔矩阵的广义Moore-Penrose逆.给出了一些广义Moore-Penrose逆存在的充要条件以及广义Moore-Penrose逆的一些刻划.     

Estimates for coefficients of univalent functions from integral means and Grunsky inequalities

Sharp bounds are obtained for the coefficients of inverses of univalent functions in the class Σ(p) by using results on integral means and generalized Grunsky inequalities. A new and elementary proof is given for a result due to Löwner about sharp bounds for coefficients of inverses of functions in the classS.     

On generalized Boolean functions i

In this paper we introduce the concept of generalized Boolean function. Such a function has its arguments and values in a Boolean algebra and can be written in a manner similar to the canonical disjunctive form, but instead of the product of simple or complemented variables, the product of values of certain functions is used. Every Boolean function is a generalized Boolean one but the converse is not true. The set of all generalized Boolean function “generated” by some fixed function is a Boolean algebra.     

M-P invertible matrices and unitary groups over Fq 2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

Generalized inverses of matrices over a finite field

For a given m × n matrix A of rank r over a finite field F, the number of generalized inverses, of reflexive generalized inverses, of normalized generalized inverses, and of pseudoinverses of A are determined by elementary methods. The more difficult problem of determining which m × n matrices A of rank r over F have normalized generalized inverses and which have pseudoinverses is solved. Moreover, the number of such matrices which possess normalized generalized inverses and the number which possess pseudoinverses are found.     

The use of generalized inverses in restricted maximum likelihood

The calculus of generalized inverses and related concepts in matrix algebra is applied to the general restricted maximum likelihood problem. Some new results on g-inverses, Kronecker products, and matrix differentials are presented. For the restricted maximum likelihood problem we obtain generalizations of the well-known results of Aitchison and Silvey [1]. We use the approach recently developed by Heijmans and Magnus [13, 14] to allow for non-i.i.d. observations. A nonlinear seemingly unrelated regressions model with possibly singular covariance matrix and linear restrictions (NLSURSR) is analyzed, and the linear expenditure system (LES) is discussed as a special case.     

关于蕴涵BCK—代数的伴随半群

我们证明了蕴涵BCK-代数的伴随半群是一个上半格;具有条件（s)的蕴涵BCK-代数的伴随半群是一个广义布尔代数。更进一步证明了有界蕴涵BCK-代数的伴随半群是一个布尔代数。     

剩余格及有界psBCK-代数成为布尔代数的充要条件

研究了一般剩余格(未必可换)与布尔代数的关系,给出剩余格成为布尔代数的一系列充要条件.同时,进一步将这些结果推广到只含有蕴涵运算的有界psBCK-代数中,证明了在一定条件下由psBCK-代数可诱导出有界格且构成布尔代数.     

Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators

The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore-Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T⁺(I+δTT⁺)^-1. Utilizing these results, we investigate the expression for the Moore-Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore-Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore-Penrose inverse to have the simplest expression T^†(I+δTT^†)^-1. The results obtained in this paper extend and improve many recent results in this area.     

The representation and approximations of outer generalized inverses

We present a unified representation theorem for the class of all outer generalized inverses of a bounded linear operator. Using this representation we develop a few specific expressions and computational procedures for the set of outer generalized inverses. The obtained result is a generalization of the well-known representation theorem of the Moore--Penrose inverse as well as a generalization of the well-known results for the Drazin inverse and the generalized inverse A_T,S ⁽²⁾. Also, as corollaries we get corresponding results for reflexive generalized inverses.     

The pasting constructions for effect algebras

The aim of this paper is to present several techniques of constructing a lattice-ordered effect algebra from a given family of lattice-ordered effect algebras, and to study the structure of finite lattice-ordered effect algebras. Firstly, we prove that any finite MV-effect algebra can be obtained by substituting the atoms of some Boolean algebra by linear MV-effect algebras. Then some conditions which can guarantee that the pasting of a family of effect algebras is an effect algebra are provided. At last, we prove that any finite lattice-ordered effect algebra E without atoms of type 2 can be obtained by substituting the atoms of some orthomodular lattice by linear MV-effect algebras. Furthermore, we give a way how to paste a lattice-ordered effect algebra from the family of MV-effect algebras.     

A refinement of Stone duality to skew Boolean algebras

We establish two theorems that refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem, we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of étale spaces over locally compact Boolean spaces whose morphisms are étale space cohomomorphisms over continuous proper maps. In the second theorem, we prove that the category of left-handed skew Boolean -algebras whose morphisms are proper skew Boolean -algebra homomorphisms is equivalent to the category of étale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective étale space cohomomorphisms over continuous proper maps.     

完备布尔代数商格的注

本文研究了完备布尔代数L的商格Q(L),建立 L上q-集构成的集合Q′(L)与Q(L)之间的同构,得出了完备布尔代数的商格仍是完备布尔代数,并给出了完备布尔代数范畴的某些性质.     

Generalized inverses and their application to applied probability problems

The main aim of this paper is to examine the applicability of generalized inverses to a wide variety of problems in applied probability where a Markov chain is present either directly or indirectly through some form of imbedding. By characterizing all generalized inverses of I—P, where P is the transition matrix of a finite irreducible discrete time Markov chain, we are able to obtain general procedures for finding stationary distributions, moments of the first passage time distributions, and asymptotic forms for the moments of the occupation-time random variables. It is shown that all known explicit methods for examining these problems can be expressed in this generalized inverse framework. More generally, in the context of a Markov renewal process setting the aforementioned problems are also examined using generalized inverses of I—P. As a special case, Markov chains in continuous time are considered, and we show that the generalized inverse technique can be applied directly to the infinitesimal generator of the process, instead of to I—P, where P is the transition matrix of the discrete time jump Markov chain.