Related papers: Circular orderability and quandles
We introduce the notion of an orbit series in a quandle. Using this notion we define four families of quandles based on finiteness conditions on their orbit series. Intuitively, the classes tOS and tOSn correspond to finitary compositions…
We completely describe good involutions of free quandles and subquandles of twisted conjugation quandles of groups, including all Alexander quandles. As an application, we enumerate good involutions of linear quandles, and we provide…
Quandles can be regarded as generalizations of symmetric spaces. In the study of symmetric spaces, the notion of flatness plays an important role. In this paper, we define the notion of flat quandles, by referring to the theory of…
We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles.
Given a nonempty set $\mathcal{L}$ of linear orders, we say that the linear order $L$ is $\mathcal{L}$-convex embeddable into the linear order $L'$ if it is possible to partition $L$ into convex sets indexed by some element of $\mathcal{L}$…
We study topological properties of circularly ordered dynamical systems and prove that every such system is representable on a Rosenthal Banach space, hence, is also tame. We derive some consequences for topological groups. We show that…
We already saw in [A1] that the space of dynamically marked rational maps can be identified to a subspace of the space of covers between trees of spheres on which there is a notion of convergence that makes it sequentially compact. In the…
Racks and quandles are algebraic structures with a single binary operation that is right self-distributive and right invertible, and additionally idempotent in the case of quandles. The invertibility condition is equivalent to the existence…
We study the space of left-orderings on groups with finitely many Conradian orderings. We show that, within this class of groups, having an isolated left-ordering is equivalent to having finitely many left-orderings.
While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and…
We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits…
A well ordering < of a topological space X is "left-separating" if $\{x'\in X: x'< x\}$ is closed in X for any x in X. A space is "left-separated" if it has a left-separating well-ordering. The left-separating type, $ord_l(X)$, of a…
In this paper, we introduce round and sleek topological spaces and study their properties.
We introduce a class of orders on $P^d$ called Geigle-Lenzing orders and show that they have tilting bundles. Moreover we show that their module categories are equivalent to the categories of coherent sheaves on Geigle-Lenzing spaces…
The class of LOTS (linearly ordered topological spaces, i.e. spaces equipped with a topology generated by a linear order) contains many important spaces, like the set of real numbers, the set of rational numbers and the ordinals. Such…
A Cech closure space $(X,u)$ is a set $X$ with a (Cech) closure operator $u$ which need not be idempotent. Many properties which hold in topological spaces hold in Cech closure spaces as well. The notions of proper (splitting) and…
This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It…
We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of…
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can…
The paper focuses on the structure of fundamental sequences of ordinals smaller than $\epsilon_0$. A first result is the construction of a monadic second-order formula identifying a given structure, whereas such a formula cannot exist for…