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Given a discrete semigroup $$(S,\cdot )$$ , there is a natural operation on the Stone–Cech compactification $$\beta S$$ of S which extends the operation of S and makes $$(\beta S,\cdot )$$ a compact right topological semigroup with S contained in its topological center. If S and T are discrete semigroups, $$p\in \beta S$$ , and $$q\in \beta T$$ , then the tensor product $$p\otimes q$$ is a member of $$\beta (S\times T)$$ . It is known that tensor products are both algebraically and topologically rare in $$\beta (S\times T)$$ . We investigate when the algebraic product of two tensor products is again a tensor product. We get a simple characterization for a large class of semigroups. The characterization is in terms of a notion of cancellation. We investigate where that notion sits among standard cancellation notions.

In this article, we establish necessary and sufficient condition on a topological Clifford semigroup to be a semilattice of topological groups. As a consequence, we show that a topological Clifford...

An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group.

Let S be a foundation topological semigroup and $$M_a(S)$$ the space of all measures $$\mu \in M(S)$$ for which the maps $$x\longmapsto |\mu |*\delta _{x}$$ and $$x\longmapsto \delta _{x}*|\mu |$$ from S into M(S) are weakly continuous. In the present paper, we introduce and study the concept of $$\phi$$ -amenability for S and investigate the relations between $$\phi$$ -amenability of S and essential $$\widehat{\phi }$$ -amenability of $$M_a(S)$$ , where $$\phi$$ is a character on S and $$\widehat{\phi }$$ is the extension of $$\phi$$ to $$M_a(S)$$ .

In this paper, we study the Raǐkov completion of invariant fuzzy metric groups and complete fuzzy metric semigroups (in the sense of Kramosil and Michael). We establish that: (1) if $ (G, M,\ast) $ is a fuzzy metric group such that $ (M,\ast) $ is invariant, then the Raǐkov completion $ \varrho G $ of $ (G,\tau_{M}) $ is a fuzzy metric group $ (\varrho G, \widetilde{M},\ast) $ such that $ (\widetilde{M},\ast) $ is invariant on $ \varrho G $ and $ \widetilde{M}_{|G\times G\times [0,\infty)} = M $; (2) if $ (G, M, \ast) $ is a fuzzy metric semigroup such that $ (M, \ast) $ is invariant, then a fuzzy metric completion $ (\widetilde{G}, \widetilde{M}, \ast) $ of $ (G, M, \ast) $ is a fuzzy metric semigroup and $ (\widetilde{M}, \ast) $ is invariant.

Let $S$ be an infinite discrete semigroup. The operation on $S$ extends uniquely to the Stone-Cech compactification $\beta S$ making $\beta S$ a compact right topological semigroup with $S$ contained in its topological center. As such, $\beta S$ has a smallest two sided ideal, $K(\beta S)$. An ultrafilter $p$ on $S$ is \emph{minimal} if and only if $p \in K(\beta S)$. We show that any minimal ultrafilter $p$ factors into a thick part and a syndetic part. That is, there exist filters $\mathcal F$ and $\mathcal G$ such that $\mathcal F$ consists only of thick sets, $\mathcal G$ consists only of syndetic sets, and $p$ is the unique ultrafilter containing $\mathcal F \cup \mathcal G$. Letting $L = \widehat{\mathcal F}$ and $C = \widehat{\mathcal G}$, the sets of ultrafilters containing $\mathcal F$ and $\mathcal G$ respectively, we have that $L$ is a minimal left ideal of $\beta S$, $C$ meets every minimal left ideal of $\beta S$ in exactly one point, and $L \cap C = \{p\}$. We show further that $K(\beta S)$ can be partitioned into relatively closed sets, each of which meets each minimal left ideal in exactly one point. With some weak cancellation assumptions on $S$, one has also that for each minimal ultrafilter $p$, $S^* \setminus \{p\}$ is not normal. In particular, if $p$ is a member of either of the disjoint sets $K(\beta \mathbb N , +)$ or $K(\beta \mathbb N , \cdot)$, then $\mathbb N^* \setminus \{p\}$ is not normal.

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.

In this paper we study submonoids of the monoid $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_\infty^{\!\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_\infty^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_\infty^{[\underline{1}]}$ of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_\infty^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups., Comment: 12 pages. arXiv admin note: text overlap with arXiv:1802.03598; and text overlap with arXiv:1908.08521 by other authors

In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.

In this paper, we show that every T 1 topological semigroup satisfying condition (Δ) can be embedded into a topological semigroup ( D , σ , ⊙ ) , where ( D , ⊑ ) is a domain. Furthermore, by considering the maximal point topological semigroup of a continuous prequantale, it is proven that every T 1 topological semigroup satisfying condition (Δ) has a continuous prequantale model, which may not be bounded complete.

For a countably infinite group $\Gamma$, let $\mathcal{W}_\Gamma$ denote the space of all weak equivalence classes of measure-preserving actions of $\Gamma$ on atomless standard probability spaces, equipped with the compact metrizable topology introduced by Ab\'{e}rt and Elek. There is a natural multiplication operation on $\mathcal{W}_\Gamma$ (induced by taking products of actions) that makes $\mathcal{W}_\Gamma$ an Abelian semigroup. Burton, Kechris, and Tamuz showed that if $\Gamma$ is amenable, then $\mathcal{W}_\Gamma$ is a topological semigroup, i.e., the product map $\mathcal{W}_\Gamma \times \mathcal{W}_\Gamma \to \mathcal{W}_\Gamma \colon (\mathfrak{a}, \mathfrak{b}) \mapsto \mathfrak{a} \times \mathfrak{b}$ is continuous. In contrast to that, we prove that if $\Gamma$ is a Zariski dense subgroup of $\mathrm{SL}_d(\mathbb{Z})$ for some $d \geqslant 2$ (for instance, if $\Gamma$ is a non-Abelian free group), then multiplication on $\mathcal{W}_\Gamma$ is discontinuous, even when restricted to the subspace $\mathcal{FW}_\Gamma$ of all free weak equivalence classes., Comment: 14 pages; v2: minor changes following a referee report

Let $\pi\colon T\times X\rightarrow X$ with phase map $(t,x)\mapsto tx$, denoted $(\pi,T,X)$, be a \textit{semiflow} on a compact Hausdorff space $X$ with phase semigroup $T$. If each $t\in T$ is onto, $(\pi,T,X)$ is called surjective; and if each $t\in T$ is 1-1 onto $(\pi,T,X)$ is called invertible and in latter case it induces $\pi^{-1}\colon X\times T\rightarrow X$ by $(x,t)\mapsto xt:=t^{-1}x$, denoted $(\pi^{-1},X,T)$. In this paper, we show that $(\pi,T,X)$ is equicontinuous surjective iff it is uniformly distal iff $(\pi^{-1},X,T)$ is equicontinuous surjective. As applications of this theorem, we also consider the minimality, distality, and sensitivity of $(\pi^{-1},X,T)$ if $(\pi,T,X)$ is invertible with these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of $\mathbb{Z}$-flow with compact zero-dimensional phase space., Comment: 61 pages

In this paper, we introduce and study the notion of quasi-multipliers on a semi-topological semigroup [Formula: see text]. The set of all quasi-multipliers on [Formula: see text] is denoted by [Formula: see text]. First, we study the problem of extension of quasi-multipliers on topological semigroups to its Stone–Čech compactification. Indeed, we prove if [Formula: see text] is a topological semigroup such that [Formula: see text] is pseudocompact, then [Formula: see text] can be regarded as a subset of [Formula: see text] Moreover, with an extra condition we describe [Formula: see text] as a quotient subsemigroup of [Formula: see text] Finally, we investigate quasi-multipliers on topological semigroups, its relationship with multipliers and give some concrete examples.

In this paper we study and develop the retracting property in homotopy theory for finite topological semigroups by introducing the notions of $DS-$retract sets and $US-$retract sets. We show that these sets are stronger forms of strong $S-$deformation retract sets. Furthermore, we study the uniqueness of retraction maps and some topological properties for these retract sets.