Factorisation in Topological Monoids

非阿基米德拓扑幺半群

Non-archimedean topological monoids

本文定义并研究左、右非阿基米德拓扑幺半群，利用 Stone 与 Pontryagin 对偶刻画其结构，并证明两类 Polish 幺半群分别在相应可分可度量化 NA 幺半群中具有泛性。

Infinitary commutativity and fundamental groups of topological monoids

The well-known Eckmann-Hilton Principle may be applied to prove that fundamental groups of $H$-spaces are commutative. In this paper, we identify an infinitary analogue of the Eckmann-Hilton Principle that applies to fundamental groups of all topological monoids and slightly more general objects called pre-$Δ$-monoids. In particular, we show that every pre-$Δ$-monoid $M$ is "transfinitely $π_1$-commutative" in the sense that permutation of the factors of any infinite loop-concatenation indexed by a countably infinite order and based at the identity $e\in M$ is a homotopy invariant action. We a…

Kuratowski monoids of $n$-topological spaces

Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set $X$ endowed with $n$ pairwise comparable topologies $τ_1\subset\dots\subsetτ_n$, by repeated application of the operations of complement and closure in the topologies $τ_1,\dots,τ_n$ to a subset $A\subset X$ we can obtains at most $2K(n)=2\sum_{i,j=0}^n\binom{i+j}{i}\binom{i+j}{j}$ distinct sets.

拓扑嵌入到变换单形中

Topological embeddings into transformation monoids

本文研究哪些拓扑半群能拓扑嵌入到全变换单形或对称逆单形中，刻画了属于特定类别的拓扑半群的特征，并构造了不嵌入到全变换单形的可数波兰拓扑半群的例子，回答了一个开放问题。

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 …

Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image

In this paper we study the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology $τ$ on $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),τ)$ is a topolo…

Topological monoids of almost monotone injective co-finite partial selfmaps of positive integers

In this paper we study the semigroup $I_\infty^\dnearrow(N)$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $I_\infty^\dnearrow(N)$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $τ$ on $I_\infty^\dnearrow(N)$ such that $(I_\infty^\dnearrow(N),τ)$ is a semitopological semigroup is discrete, describe the closure of $(I_\infty^\dnearrow(N),τ)$ in a…

Pseudocompactness, products and topological Brandt $λ^0$-extensions of semitopological monoids

In the paper we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, $ω$-boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) to\-pological Brandt $λ_i^0$-extensions of semitopological monoids with zero. In particular we show that if $\big\{ \big(B^0_{λ_i}(S_i),τ^0_{B(S_i)}\big) \colon i\in\mathscr{I}\big\}$ is a family of Hausdorff pseudocompact to\-pological Brandt $λ_i^0$-extensions of pseudocompact semitopological monoids with zero such…

On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise

In the paper we study algebraic properties of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ of cofinite partial isometries of the set of positive integers $\mathbb{N}$ with the bounded finite noise $j$. For the monoids $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ we prove counterparts of some classical results of Eberhart and Selden describing the closure of the bicyclic semigroup in a locally compact topological inverse semigroup. In particular we show that for any positive integer $j$ every Hausdorff shift-continuous topology $τ$ on $\mathbf{I}\mathbb{N}_{\infty}^{\bo…

On monoids of injective partial selfmaps almost everywhere the identity

In this paper we study the semigroup $\mathscr{I}^{\infty}_λ$ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality $λ$. We describe the Green relations on $\mathscr{I}^{\infty}_λ$, all (two-sided) ideals and all congruences of the semigroup $\mathscr{I}^{\infty}_λ$. We prove that every Hausdorff hereditary Baire topology $τ$ on $\mathscr{I}^{\infty}_ω$ such that $(\mathscr{I}^{\infty}_ω,τ)$ is a semitopological semigroup is discrete and describe the closure of the discrete semigroup $\mathscr{I}^{\infty}_λ$ in a topological semigroup. Also we show that …

$λ$-Semidirect Products of Inverse Monoids are Weakly Schreier Extensions

A split extension of monoids with kernel k: N -> G, cokernel e: G -> H and splitting s: H -> G is weakly Schreier if each element g in G can be written g = k(n)se(g) for some n in N. The characterization of weakly Schreier extensions allows them to be viewed as something akin to a weak semidirect product. The motivating examples of such extensions are the Artin glueings of topological spaces and, of course, the Schreier extensions of monoids which they generalise. In this paper we show that the lambda-semidirect products of inverse monoids are also examples of weakly Schreier extensions. The c…

Kuratowski幺半群的边界-边缘扩张

Boundary-Border Extensions of the Kuratowski Monoid

该研究探讨Kuratowski幺半群的边界扩张GE幺半群的基数规律，定义了新型拓扑空间Kuratowski disconnected，分析了GE幺半群的70种坍缩方式与自身的依赖关系，解决了两个公开问题，计算机实验起到关键作用。