P-析取强P-反演半群

介绍了强P-反演半群的概念,刻划了P-析取强P-反演半群的某些特征.     

P-反演半群的P-次直积

本文研究了P-反演半群的P-次直积和E-囿P-反演盖.利用P-反演半群的P-次直积的结构,引入了P-反演半群的E-囿P-反演盖概念,并刻画了它们的结构.最后,讨论了P-反演半群到给定群上的囿P-次同态,推广了文献[9]中的一些结果到P-反演半群上.     

P-正则半群上的P-同余

给出了P-正则半群内的P-核正规系统的一个等价刻划并用P-核正规系统描速了P-正则半群上的强P-同余。     

P-反演半群上的正则P-同余

令S(P)为P-反演半群，本文借助于P-核正规系来刻画S(P)上的强P-同余，证明了S(P)上的任一正则P-同余可以决定S(P)的一个P-核正规系；反之，S(P)的任一P-核正规系可以决定S(P)上的一个正则P-同余．     

Γ-环的P-根、次P-根与拟P-根

在Γ-环中定义P-根，次P-根与拟P-根的概念，讨论它们的性质及相互间的关系．给出了次P-根的构造，证明了对Γ-环的任一代数性质P，总可确定两个Amitsur-Kurosh根．同时，对Γ-环的几个具体报的研究做了统一，拓广了Γ-环报理论的研究领域．     

P-反演半群上的强P-同余

本文介绍了P-反演半群S(P)的概念，借助于核与迹刻画了P-反演半群上的强P-同余，并且证明了S(P)上的任一强P-同余可以决定S(P)的一个强P-同余对，反之S(P)的任一强P-同余对，可以决定S(P)上的一个强P-同余.     

弱P-反演半群上的强P-同余

本文利用核-迹方法，研究了弱P-反演半群上的强P-同余．给出了强P-同余对和强P-同余关系之间的结构定理．     

P-可测函数的构造性质

可测函数的构造性质是定义它关于测度μ的积分的理论基础.为了在P-测度空间上定义P-积分,借鉴可测函数的构造性质,引入了P-示性函数、P-简单函数、P-初等函数以及P-可测函数的概念,在此基础上系统地研究了P-实可测函数、有界P-实可测函数和非负P-可测函数与P-简单函数序列及P-初等函数序列的收敛关系;找出了P-实可测的充分必要条件;证明了实P-可测函数正部和负部都是非负P-实可测函数,最终得出任何P-实可测函数均可以表示为二非负P-可测函数之差,为定义P-积分提供了理论依据.     

弱P-反演半群上的强P-同余Ⅱ

刻画了弱P-反演半群S(P)上满足P-trμN=πN的最大的强P-同余;给出了S(P)上的特征核正规系的一种抽象刻画.     

P-正则半群的结构

本文定义了一类称之为PR-系的代数,证明了每个PR-系是一个P-正则半群, 而且每个P-正则半群可用这种方式来进行构造.     

P-反演半群上强P-同余的P-核正规系

Let S(P) be a P-inversive semigroup. In this paper we describe the strong P-congruences on S(P) in terms of their P-kernel normal systems. We prove that any strong P-congruence on S(P) can present a P-kernel normal system; conversely any P-kernel normal system of S(P) can determine a strong P-congruence.     

E-反演P-半群上的强P-同余

令S(P)为E-反演P-半群．本文用核与迹研究了S(P)上的强P-同余．证明了S(P)的任一强P-同余对，可以决定S(P)上的一个强P-同余；反之，S(P)上的任一强P-同余，可以决定S(P)的一个强P-同余对。     

Sublattices of the Lattices of Strong P-Congruences on P-Inversive Semigroups

In this paper, we describe strong P-congruences and sublattice-structure of the strong P-congruence lattice C_P(S) of a P-inversive semigroup S(P). It is proved that the set of all strong P-congruences C_P(S) on S(P) is a complete lattice. A close link is discovered between the class of P-inversive semigroups and the well-known class of regular ⋆-semigroups. Further, we introduce concepts of strong normal partition/equivalence, C-trace/kernel and discuss some sublattices of C_P(S). It is proved that the set of strong P-congruences, which have C-traces (C-kernels) equal to a given strong normal equivalence of P (C-kernel), is a complete sublattice of C_P(S). It is also proved that the sublattices determined by C-trace-equaling relation θ and C-kernel-equaling relation κ, respectively, are complete sublattices of C_P(S) and the greatest elements of these sublattices are given.     

Attractors and dimension of dissipative lattice systems

In this paper, by using the argument in [Q.F. Ma, S.H. Wang, C.K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51(6) (2002), 1541-1559.], we prove that the condition given in [S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations 200 (2004) 342-368.] for the existence of a global attractor for the semigroup associated with general lattice systems on a discrete Hilbert space is a sufficient and necessary condition. As an application, we consider the existence of a global attractor for a second-order lattice system in a discrete weighted space containing all bounded sequences. Finally, we show that the global attractor for first-order and partly dissipative lattice systems corresponding to (partly dissipative) reaction-diffusion equations and second-order dissipative lattice systems corresponding to the strongly damped wave equations have finite fractal dimension if the derivative of the nonlinear term is small at the origin.     

Partial Kernel Normal Systems for Eventually Regular Semigroups

The set of P-partial kernel normal systems for an eventually regular semigroup S forms a complete lattice, which is a completely ∧-homomorphic image of C(S). Every regular congruence on S is uniquely determined by its P-partial kernel normal system.     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

Searching for extensible Korobov rules

Extensible lattice sequences have been proposed and studied in [F.J. Hickernell, H.S. Hong, Computing multivariate normal probabilities using rank-1 lattice sequences, in: G.H. Golub, S.H. Lui, F.T. Luk, R.J. Plemmons (Eds.), Proceedings of the Workshop on Scientific Computing (Hong Kong), Singapore, Springer, Berlin, 1997, pp. 209–215; F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138; F.J. Hickernell, H.Niederreiter, The existence of good extensible rank-1 lattices, J. Complexity 19 (2003) 286–300]. For the special case of extensible Korobov sequences, parameters can be found in [F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C.Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138]. The searches made to obtain these parameters were based on quality measures that look at several projections of the lattice. Because it is often the case in practice that low-dimensional projections are very important, it is of interest to find parameters for these sequences based on measures that look more closely at these projections. In this paper, we prove the existence of “good” extensible Korobov rules with respect to a quality measure that considers two-dimensional projections. We also report results of experiments made on different problems where the newly obtained parameters compare favorably with those given in [F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138].