Tiling Groupoids and Bratteli Diagrams

Let T be an aperiodic and repetitive tiling of ${{\mathbb R}^d}$ with finite local complexity. Let Ω be its tiling space with canonical transversal ${\Xi}$ . The tiling equivalence relation ${R_\Xi}$ is the set of pairs of tilings in ${\Xi}$ which are translates of each others, with a certain (étale) topology. In this paper ${R_\Xi}$ is reconstructed as a generalized “tail equivalence” on a Bratteli diagram, with its standard AF -relation as a subequivalence relation. Using a generalization of the Anderson–Putnam complex (Bellissard et al. in Commun. Math. Phys. 261:1–41, 2006) Ω is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram ${{\mathcal B}}$ is built from this sequence, and its set of infinite paths ${\partial {\mathcal B}}$ is homeomorphic to ${\Xi}$ . The diagram ${{\mathcal B}}$ is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an étale equivalence relation ${R_{\mathcal B}}$ on ${\partial {\mathcal B}}$ which is homeomorphic to ${R_\Xi}$ , and contains the AF-relation of “tail equivalence”.     

关于Poisson群胚的结构

令（ΓP,α,β）是Poisson群胚．如果它的每个α-纤维与β-纤维至多交于一点,则Γ在任一点x的特征分布有直和分解△（x）=△α（x）+△β（x）,其中△α（x）Txα－1（u）,△β（x）Txβ-1（u）且它们都是△（x）的辛子空间．由此得到辛叶Sx的辛子流形S和S,使在映射α之下,S辛微分同胚于P中辛叶Su,在映射β之下,S反辛微分同胚于P中辛叶Sv（定理4和5）．对于一般的Poisson群胚,也可得到类似的S和S,它们差一局部辛微分同胚是唯一确定的（定理6）．把以上结果用于辛群胚,还可得到一些更具体的性质（定理7及其推论）．     

辛群胚的弱Morita等价与orbifold基本群

本文研究辛orbifold群胚的弱Morita等价,证明了两个辛orbifold群胚弱Morita等价当且仅当其orbifold基本群同构.     

Finitary Orbit Equivalence of Odometers

A finitary orbit equivalence mapping between the binary andternary odometers is constructed. 2000 Mathematics Subject Classification28D05, 37A05, 37A20.     

等价关系的判定及性质

设R是集合A上的二元关系,要判定R在A上是否是等价关系,一般来讲,只能从定义出发,当R包含的序偶较多时,从定义出发又比较难于判定.从关系矩阵出发,给出一种判定方法,并讨论等价关系的矩阵性质.     

Orbit Equivalence and Topological Conjugacy of Affine Actionson Compact Abelian Groups

 We study topological rigidity of affine actions on compact connected metrizable abelian groups. We also classify one parameter flows of translations up to orbit equivalence and discrete group actions by translations up to topological conjugacy. (Received 21 December 1998; in revised form 2 June 1999)     

辛群胚与泊松群胚作用的充要条件

研究了辛群胚与泊松群胚的作用.利用李群胚作用及相关性质,得到了李群胚作用成为辛群胚和泊松群胚作用的充要条件,推广了辛群胚和泊松群胚的性质,为辛群胚与泊松群胚理论的进一步研究起到了推动作用.     

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.     

Semigroups of Transformations Preserving an Equivalence Relation and a Cross-Section

Abstract

For a set X, an equivalence relation ρ on X, and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T(X, ρ, R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y)?∈?ρ then (xa, ya)?∈?ρ) and R (if r?∈?R then ra?∈?R). The semigroup T(X, ρ, R) is the centralizer of the idempotent transformation with kernel ρ and image R. We determine the structure of T(X, ρ, R) in terms of Green's relations, describe the regular elements of T(X, ρ, R), and determine the following classes of the semigroups T(X, ρ, R): regular, abundant, inverse, and completely regular.     

Equivalence of Room designs. II

The relation of equivalence of Room designs is investigated with respect to the class of Room designs formed from starters with adders in the conventional way. It is shown that, if p^k = 3 + 4t, $\text{s = t ? 2[}\text{t}\text{2}\text{]}$, and $\text{I = (}\text{1}\text{k}\text{) ∑}{}_{\mathrm{<mtext>d</mtext><mtext>k</mtext>}}\text{P}{}^{\mathrm{d}}\text{φ}$ ($\text{k}\text{d}$), then it is possible to construct, without duplication, at least $\text{(I + (?3)}{}^{\mathrm{1–8}}\text{)}\text{8}$ inequivalent patterned Room designs of side p^k. This result implies, for instance, the existence of at least 11 equivalence classes of Room designs of side 87 which contain a patterned Room design; it also implies the existence of at least 5 · 10⁵ equivalence classes of Room designs of side 7⁹ which contain a patterned Room design. Some additional results are obtained.     

铺瓷砖问题的推广

铺瓷砖问题是数学模型中著名的问题,在研究了铺瓷砖问题的基础上,对其进行了推广,并给出了各种推广情形的结果.     

The Endotopism Semigroups of a Partial Equivalence Relation

Siberian Mathematical Journal - We study the endotopism semigroups of a&nbsp;nonreflexive partial equivalence. We&nbsp;describe the existence conditions for six types of endotopisms and...     

Natural Tiling, Lattice Tiling and Lebesgue Measure of Integral Self-Affine Tiles

In the existing theory of self-affine tiles, one knows thatthe Lebesgue measure of any integral self-affine tile correspondingto a standard digit set must be a positive integer and everyintegral self-affine tile admits some lattice Zⁿ as a translationtiling set of Rⁿ. In this paper, we give algorithms to evaluatethe Lebesgue measure of any such integral self-affine tile Kand to determine all of the lattice tilings of Rⁿ by K. Moreover,we also propose and determine algorithmically another type oftranslation tiling of Rⁿ by K, which we call natural tiling.We also provide an algorithm to decide whether or not Lebesguemeasure of the set K (K+j), j–Zⁿ, is strictly positive.     

基于Fuzzy等价关系的矩形聚类方法

证明了Boole矩阵与模糊矩阵等价的充分必要条件,总结出Fuzzy聚类的矩形方法.针对《中国大学评价》中的6所林业院校,应用矩形方法,直接从模糊关系矩阵逐步降低阈值,得到分类的矩形表.此方法操作简单,形象直观,分类合理.     

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.     

Fibrations of Ordered Groupoids and the Factorization of Ordered Functors

We investigate canonical factorizations of ordered functors of ordered groupoids through star-surjective functors. Our main construction is a quotient ordered groupoid, depending on an ordered version of the notion of normal subgroupoid, that results in the factorization of an ordered functor as a star-surjective functor followed by a star-injective functor. Any star-injective functor possesses a universal factorization through a covering, by Ehresmann’s Maximum Enlargement Theorem. We also show that any ordered functor has a canonical factorization through a functor with the ordered homotopy lifting property.     

On the 1-Homotopy Type of Lie Groupoids

We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbifold maps.