Compactifications of Semigroups and Semigroup Actions

On embeddings of proper and equicontinuous actions in zero-dimensional compactifications

We provide a tool for studying properly discontinuous actions of non-compact groups on locally compact, connected and paracompact spaces, by embedding such an action in a suitable zero-dimensional compactification of the underlying space with pleasant properties. Precisely, given such an action $(G,X)$ we construct a zero-dimensional compactification $μX$ of $X$ with the properties: (a) there exists an extension of the action on $μX$, (b) if $μL\subseteq μX\setminus X$ is the set of the limit points of the orbits of the initial action in $μX$, then the restricted action $(G,μX\setminus μL)$ re…

Metric vs topological receptive entropy of semigroup actions

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov. We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowe…

A note on sensitivity of semigroup actions

It is well known that for a transitive dynamical system (X,f) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey, Silverman, and also by Glasner and Weiss. In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X,f) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in a recent book of Glasner. Our aim is to gene…

Pseudocompact primitive topological inverse semigroups

In the paper we study pseudocompact primitive topological inverse semigroups. We describe the structure of pseudocompact primitive topological inverse semigroups and show that a Tychonoff product of a family of pseudocompact primitive topological inverse semigroups is a pseudocompact topological space. Also we prove that the Stone-Čech compactification of a pseudocompact primitive topological inverse semigroup is a compact primitive topological inverse semigroup.

论构造拓扑群紧化的两种方法

On two methods of constructing compactifications of topological groups

本文研究拓扑群紧化按代数运算延拓性的分类，比较 Ellis 方法与超空间图像表示法，分别得到左乘连续延拓及对合与左乘共同延拓的紧化。

全不连通半群紧化：闭开集完全布尔代数的非内省性

Totally Disconnected Semigroup Compactifications: Non-Introversion of the Full Boolean Algebra of Clopen Sets

通过单个闭开集和两个相关网刻画拓扑群G的闭开集完全布尔代数B(G)左内省的条件，证明第一可数σ-紧全不连通局部紧群G中B(G)左内省当且仅当G紧或离散，肯定回答Stephens和Stokke的问题。

A topological correspondence between partial actions of groups and inverse semigroup actions

We present some generalizations of the well-known correspondence, found by R. Exel, between partial actions of a group $G$ on a set $X$ and semigroup homomorphism of $S(G)$ on the semigroup $I(X)$ of partial bijections of $X,$ being $S(G)$ an inverse monoid introduced by Exel. We show that any unital premorphism $θ:G\to S$, where $S$ is an inverse monoid, can be extended to a semigroup homomorphism $θ^*:T\to S$ for any inverse semigroup $T$ with $S(G)\subseteq T\subseteq P^*(G)\times G,$ being $P^*(G)$ the semigroup of non-empty subset of $G$, and such that $E(S)$ satisfies some lattice theore…

Semigroup Closures of Finite Rank Symmetric Inverse Semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations $\mathscr{I}_λ^n$ of the rank $\leqslant n$ is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.

Openly factorizable spaces and compact extensions of topological semigroups

We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its the Stone-Čech compactification $βS$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:S\to Y$ to a second countable space $Y$ can be written as the composition $f=g\circ p$ of an open map $p:X\to Z$ onto a second countable space $Z$ and a map $g:Z\to Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.

The maximal G-compactifications of G-spaces with special actions

An action on a G-space induces uniformities on the phase space. It is shown when the maximal G-compactification of a G-space can be obtained as a completion of the phase space with respect to one of these uniformities. Structure of G-spaces with special actions is investigated.

On feebly compact inverse primitive (semi)topological semigroups

We study the structure of inverse primitive feebly compact semitopological and topological semigroups. We find conditions when the maximal subgroup of an inverse primitive feebly compact semitopological semigroup $S$ is a closed subset of $S$ and describe the topological structure of such semiregular semitopological semigroups. Later we describe the structure of feebly compact topological Brandt $λ^0$-extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In particular we show that inversion in a quasi-regular primitive inverse feebly com…

乘积空间上连续函数的延拓、Bohr紧化与殆周期函数

Extension of continuous functions on product spaces, Bohr Compactification and Almost Periodic Functions

本文刻画了乘积空间上可连续延拓到其紧化的连续函数，并由此将任意拓扑结构的Bohr紧化实现为殆周期函数所成交换Banach代数的Gelfand空间，同时应用于殆周期函数空间之间等距映射的表示。

关于无限线性有序集有界秩有限偏序同构半群的紧致拓扑

On compact topologies on the semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set

本文研究了无限线性有序集有界秩的有限偏序同构半群的紧致拓扑性质，证明了其存在唯一的紧致拓扑，并揭示了其博尔紧致化的平凡性。

Compactifications of topological groups

Every topological group $G$ has some natural compactifications which can be a useful tool of studying $G$. We discuss the following constructions: (1) the greatest ambit $S(G)$ is the compactification corresponding to the algebra of all right uniformly continuous bounded functions on $G$; (2) the Roelcke compactification $R(G)$ corresponds to the algebra of functions which are both left and right uniformly continuous; (3) the weakly almost periodic compactification $W(G)$ is the envelopping compact semitopological semigroup of $G$ (`semitopological' means that the multiplication is separately …

On a semitopological polycyclic monoid

We study algebraic structure of the $λ$-polycyclic monoid $P_λ$ and its topologizations. We show that the $λ$-polycyclic monoid for an infinite cardinal $λ\geqslant 2$ has similar algebraic properties so has the polycyclic monoid $P_n$ with finitely many $n\geqslant 2$ generators. In particular we prove that for every infinite cardinal $λ$ the polycyclic monoid $P_λ$ is a congruence-free combinatorial $0$-bisimple $0$-$E$-unitary inverse semigroup. Also we show that every non-zero element $x$ is an isolated point in $(P_λ,τ)$ for every Hausdorff topology $τ$ on $P_λ$, such that $(P_λ,τ)$ is a …