Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones

In this paper we explore the extent to which the algebraic structure of a monoid $M$ determines the topologies on $M$ that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids. If $M$ is a topological monoid such that every homomorphism from $M$ to a second countable topological monoid $N$ is continuous, then we say that $M$ has \emph{automatic continuity}. We show that many well-known monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid $\mathbb{N}^\mathbb{N}$; the full binary relation monoid $B_{\mathbb{N}}$; the partial transformation monoid $P_{\mathbb{N}}$; the symmetric inverse monoid $I_{\mathbb{N}}$; the monoid Inj$(\mathbb{N})$ consisting of the injective functions on $\mathbb{N}$; and the monoid $C(2^{\mathbb{N}})$ of continuous functions on the Cantor set. We show that the pointwise topology on $\mathbb{N}^\mathbb{N}$, and its analogue on $P_{\mathbb{N}}$, are the unique Polish semigroup topologies on these monoids. The compact-open topology is the unique Polish semigroup topology on $C(2^\mathbb{N})$ and $C([0, 1]^\mathbb{N})$. There are at least 3 Polish semigroup topologies on $I_{\mathbb{N}}$, but a unique Polish inverse semigroup topology. There are no Polish semigroup topologies $B_{\mathbb{N}}$ nor on the partitions monoids. At the other extreme, Inj$(\mathbb{N})$ and the monoid Surj$(\mathbb{N})$ of all surjective functions on $\mathbb{N}$ each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on $\mathbb{N}^\mathbb{N}$, $P_{\mathbb{N}}$, and Inj$(\mathbb{N})$ coincide with the pointwise topology; and we characterise the Zariski topology on $B_{\mathbb{N}}$. In Section 7: clones.Comment: 51 pages (Section 7 about clones was added in version 4

Topological Graph Inverse Semigroups

To every directed graph $E$ one can associate a \emph{graph inverse semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$. These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, Cuntz-Krieger $C^*$-algebras, and Toeplitz $C^*$-algebras. We investigate topologies that turn $G(E)$ into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, $G(E)\setminus \{0\}$ must be discrete for any directed graph $E$. On the other hand, $G(E)$ need not be discrete in a Hausdorff semigroup topology, and for certain graphs $E$, $G(E)$ admits a $T_1$ semigroup topology in which $G(E)\setminus \{0\}$ is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of $G(E)$ in larger topological semigroups.Peer reviewe

Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances

Let ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U≈V if there exists a finite subset F of ΩΩ such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω. The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in ΩΩ where d is the minimum cardinality of a dominating family for NN. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in ΩΩ are 0, 1, 2, and d. We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2ℵ0.PostprintPeer reviewe

Kingianins O-Q: Pentacyclic polyketides from Endiandra kingiana as inhibitor of Mcl-1/Bid interaction

A phytochemical study of the EtOAc-soluble part of the methanolic extract of the bark of Endiandra kingiana led to the isolation of three new pentacyclic kingianins as racemic mixtures, kingianins O-Q (1-3), together with the known kingianins A, F, K, L, M and N (4-9), respectively. The structures of the new kingianins 1-3 were determined by 1D and 2D NMR analysis in combination with HRESIMS experiments. Kingianins A-Q were assayed for Mcl-1 binding affinity. Kingianins G and H were found to be potent inhibitors of Mcl-1/Bid interaction. A structure-activity relationship study showed that potency is very sensitive to the substitution pattern on the pentacyclic core. In addition, in contrast with the binding affinity for Bcl-xL, the levorotatory enantiomers of kingianins G, H and J exhibited similar binding affinities for Mcl-1 than their dextrorotatory counterparts, indicating that the two anti-apoptotic proteins have slightly different binding profiles

Generating sequences of functions,

J. D. Mitchell, Y. Peresse, and M. R. Quick, ;Generating sequences of functions', The Quarterly Journal of Mathematics, Vol. 58 (1): 71-79, July 2006, available online at doi: https://doi.org/10.1093/qmath/hal011. © 2006. Published by Oxford University Press.We consider the problem of obtaining an arbitrary countable collection of functions with specific properties as a composition of finitely many functions with the same property. The functions investigated are continuous, Baire-n, Lebesgue or Borel measurable, increasing, and differentiable functions on [0, 1], and increasing functions on ℕ.Peer reviewe

Cytotoxic Prenylated Stilbenes Isolated from Macaranga tanarius

With the aim of discovering new cytotoxic prenylated stilbenes of the schweinfurthin series, Macaranga tanarius was selected for detailed phytochemical investigation among 21 Macaranga species examined by using a molecular networking approach. From an ethanol extract of the fruits, seven new prenylated stilbenes, schweinfurthins K-Q (7-13), were isolated, along with vedelianin (1), schwenfurthins E-G (2-4), mappain (5), and methyl-mappain (6). The structures of the new compounds were established by spectroscopic data analysis. The relative configurations of compounds 8, 12, and 13 were determined based on ROESY NMR spectroscopic analysis. The cytotoxic activities of compounds 1-13 were evaluated against the human glioblastoma (U87) and lung (A549) cancer cell lines