保持序和等价关系的自然偏序变换半群

设X为一个集合,■_X为X上的全变换半群.设E是X上的一个等价关系,定义T_E(X)={f∈■_X:■(x,y)∈E,(f(x),f(y))∈E},则T_E(X)是由等价关系E所确定的■_X的子半群.本文中,所考虑的集合X是一个有限全序集,同时E是非平凡的且所有的E-类都是凸集.显然■_E(X)={f∈T_E(X):■_x,y∈X,x≤y蕴涵f(x)≤f(y)}是T_E(X)的一个子半群.我们赋予■_E(X)自然偏序并讨论何时■_E(X)中的两个元素是关于这个偏序是相关的,然后确定■_E(X)中那些关于≤是相容的元素.此外,还描述了极大(极小)元和覆盖元.     

一类广义变换半群的格林关系

设X是一个全序集,E是X上的一个凸等价关系．令 OE（X）={f∈TE（X）：Ax,y∈X,x≤y→f（x）≤f（y））, 其中TE（X）是E-保持变换半群．对于取定的θ∈OE（X）,在OE（X）上定义运算fog=fθg,使OE（X）成为广义半群OE（X;θ）．对于有限全序集X上的凸等价关系E,本文刻画了广义半群OE（X;θ）的正则元,描述了这个半群的格林关系．     

保持一个等价关系的部分变换半群

设X是一个集合,|X|≥3. Px为集合X上所有部分变换构成的半群.设E是集合X的一个等价关系.定义 PE(X)=｛f∈Px:(A)x,y∈domf,(x,y)∈E(→)f(x),f(y)∈E｝ 则PE(X)作成PX的一个子半群.本文讨论半群PE(X)的格林关系和正则性,并研究当等价关系E满足什么条件时,半群PE(X)是富足半群.     

关于反射等价关系的变换半群的注记

设TX是非空集合X上全变换半群,E是X上非平凡的等价关系,则T?(X)是TX的子半群.在赋予半群T?(X)自然偏序关系的条件下,本文刻画了它的相容元.     

g-拟凸函数的一个判别准则

证明了如下结果:设g∶H→H,C H是非空开的g-凸集,g(C)是凸集,f是C上的上半连续函数且存在α∈(0,1),使得f(αg(x)+(1-α)g(y))m ax{f。g(x),f。g(y)},x,y∈C,则f为C上的g-拟凸函数.     

Banach空间中非扩张映射的轨道

本文对Edelstein的问题给出一个肯定的回答。 设X是有限维Banach空间,E是X的非空子集,并设f:E→E是非扩张映射。如果存在x∈E,使得序列{f~n(x)}在X中有聚点,则对每个y∈E,轨道{f~n(y)}有界。 同时,得到了几个关于不动点的结果。     

THE ALEKSANDROV PROBLEM FOR UNIT DISTANCE PRESERVING MAPPING

1 IntroductionLet X and Y be two real metric spaces. A mappillg f: X ~ Y is called an isometryj ifd(f(x), f(y)) = d(x, y) for all x, y E X.A mapping f: X - Y satisfies the distance one preserving property (DOPP) if f for allx, y E X with d(x, y) = 1 it follows that d(f(x), f(y)) = 1.A mapping f: X ~ Y satisfies the strong distance one preserving property (SDOPP) ifffor all x, y E X with d(x, y) = 1 it follows that d(f(x), f(y)) = 1 and conversely.Problem(P) Let f: X - Y be a mappin…     

I_(E*)(X)中E类方向保序变换半群的秩

设X为有限集合,E为X上的等价关系.令OPPI_E*(X)为所有I_R*(X)中E类方向保序变换构成的半群.在一定的条件下讨论了OPPI_E*(X)的Green关系与秩.     

E-凸函数和E-拟凸函数的等价条件

在更弱的连续假设下研究集合A_(x,y)={λ∈[0,1]|f(λE(x)+(1-λ)E(y))≤λf(E(x))+(1-λ)f(E(y))}和集合A′_(x,y)={λ∈[0,1]|f(λE(x)+(1-λ)E(y))≤max{f(E(x)),f(E(y))}}的稠密性、闭性、(弱)近似凸性,得到E-凸函数和E-拟凸函数的等价条件.     

图的1-因子、f-因子和(g,f)-因子

设G是一个图且有一个1-因子F,g和f是定义在V（G）上的非负整数值函数且对每个X∈V(G)有g（X）＜f(X)≤dG（x）,且f(v(G))为偶数．（i）若对每个xy∈F有f(x)=f(y)且G－{x,y}有一个（g,f)-因子,则G有一个（g,f)-因子;（ii）若对每个xy∈F有f(X）=f（y）且G-{X,y}有f-因子,则G有f-因子.     

Green's Equivalences on Semigroups of Transformations Preserving Order and an Equivalence Relation

Let ${\cal T}_X$ be the full transformation semigroup on the set $X$, \[ T_{E}(X)=\{f\in {\cal T}_X\colon \ \forall(a,b)\in E,(f(a),f(b))\in E\} \] be the subsemigroup of ${\cal T}_X$ determined by an equivalence $E$ on $X$. In this paper the set $X$ under consideration is a totally ordered set with $mn$ points where $m\geq 2$ and $n\geq 3$. The equivalence $E$ has $m$ classes each of which contains $n$ consecutive points. The set of all order preserving transformations in $T_{E}(X)$ forms a subsemigroup of $T_E(X)$ denoted by \[ {\cal O}_{E}(X)=\{f\in T_{E}(X)\colon \ \forall\, x, y\in X, \ x\leq y \mbox{ implies } f(x)\leq f(y)\}. \] The nature of regular elements in ${\cal O}_{E}(X)$ is described and the Green's equivalences on ${\cal O}_{E}(X)$ are characterized completely.     

保持两个等价关系的变换半群的Green关系

Let Tx be the full transformation semigroup on a set X. For a non-trivial equivalence F on X, let
TF（X） = {f ∈ Tx ： arbieary （x, y） ∈ F, （f（x）,f（y）） ∈ F}.
Then TF（X） is a subsemigroup of Tx. Let E be another equivalence on X and TFE（X） = TF（X） ∩ TE（X）. In this paper, under the assumption that the two equivalences F and E are comparable and E lohtain in F, we describe the regular elements and characterize Green＇s relations for the semigroup TFE（X）.     

On the Rank of the Semigroup TE(X)

${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5.     

Regularity and Green's Relations for Semigroups of Transformations Preserving Orientation and an Equivalence

Let be the full transformation semigroup on a set X. For a non-trivial equivalence E on X, let

齐型空间上的Toeplitz型算子

设X是齐型空间.设T_(j,1)和T_(j,2)是具有非光滑核的奇异积分算子,或者是±II(I是恒等算子).令Toeplitz型算子T_b=■T_(j,1)M_T_(j,2),其中M_bf(x)=b(x)f(x).研究了当b∈BMO(X)时,T_b(f)在加权情况下的有界性,以及当b∈BMO(X)时,与经典Carderon-Zygmund算子相联的T_b(f)在Morrey空间上的有界性.     

Ultrafilters on the collection of finite subsets of an infinite set

Let $J$ be an infinite set and let $I={\cal P}_{f}( J)$, i.e., $I$ is the collection of all non empty finite subsets of $J$. Let $\beta I$ denote the collection of all ultrafilters on the set $I$. In this paper, we consider $( \beta I,\uplus ),$ the compact (Hausdorff) right topological semigroup that is the {\it Stone-$\check{C}\!\!$ech} $Compactification$ of the semigroup $\left( I,\cup \right)$ equipped with the discrete topology. It is shown that there is an injective map $A\rightarrow \beta _{A}( I) $ of ${\cal P}( J) $ into ${\cal P}( \beta I) $ such that each $\beta _{A}( I) $ is a closed subsemigroup of $ ( \beta I,\uplus ) $, the set $\beta _{J}( I) $ is a closed ideal of $( \beta I,\uplus ) $and the collection $\{ \beta _{A}( I) \mid A\in {\cal P} ( J) \} $ is a partition of $\beta I$. The algebraic structure of $\beta I$ is explored. In particular, it is shown that {\bf (1)} $\beta _{J}\left( I\right) =\overline{K( \beta I) }$, i.e., $\beta _{J}( I) $is the closure of the smallest ideal of $\beta I$, and {\bf (2)} for each non empty $A\subset J$, the set ${\cal V}_{A}=\tbigcup \{ \beta_{B}( I) \mid B\subset A\} $is a closed subsemigroup of $( \beta I,\uplus ) ,$ $\beta _{A}( I) $ is a proper ideal of ${\cal V}_{A},$ and ${\cal V}_{A}$ is the largest subsemigroup of $( \beta I,\uplus ) $ that has $ \beta _{A}( I) $ as an ideal.     

纯无限单的C*-代数通过某些C*-代数扩张的非稳定K-理论

设0→B■E■A→0是有单位元C~*-代数E的一个扩张,其中A是有单位元纯无限单的C~*-代数,B是E的闭理想.当B是E的本性理想并且同时是单的、可分的而且具有实秩零及性质(PC)时,证明了K_0(E)={[p]| p是E\B中的投影};当B是稳定C~*-代数时,证明了对任意紧的Hausdorff空间X,有■(C(X,E))/■_0(C(X,E))≌K_1(C(X,E)).     

$C(X)$上的开点与紧开拓扑

In this note we define a new topology on C(X),the set of all real-valued continuous functions on a Tychonoff space X.The new topology on C(X) is the topology having subbase open sets of both kinds:[f,C,ε[={g E C(X):|f(x)-g(x)| ε for every x∈C} and[U,r]~-={g∈C(X):g~(-1)(r)∩U≠φ},where f∈C(X),C∈KC(X)={nonempty compact subsets of X},ε 0,while U is an open subset of X and r∈R.The space C(X) equipped with the new topology T_(kh) which is stated above is denoted by C_(kh)(X).Denote X_0={x∈X:x is an isolated point of X} and X_c={x∈X:x has a compact neighborhood in X}.We show that if X is a Tychonoff space such that X_0=X_c,then the following statements are equivalent:(1) X_0 is G_δ-dense in X;(2) C_(kh)(X) is regular;(3) C_(kh)(X) is Tychonoff;(4) C_(kh)(X) is a topological group.We also show that if X is a Tychonoff space such that X_0=X_c and C_(kh)(X) is regular space with countable pseudocharacter,then X is σ-compact.If X is a metrizable hemicompact countable space,then C_(kh)(X) is first countable.