Related papers: Order separability
This article provides a brief introductory account of the theory of aperiodic order.
We provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite…
Let $G$ be a group and $G_0 \subseteq G$ be a subset. A sequence over $G_0$ means a finite sequence of terms from $G_0$, where the order of elements is disregarded and the repetition of elements is allowed. A product-one sequence is a…
A joint characterisation of the observability and controllability of a particular kind of discrete system has been developed. The key idea of the procedure can be reduced to a correct choice of the sampling sequence. This freedom, owing to…
It is well known that free independence is equivalent to the vanishing of mixed free cumulants. The purpose of this short note is to build free products of $*$-probability spaces using this as the definition of freeness and relying on free…
We prove that the common theory of nonabelian free groups has the dimensional order property, or DOP, implying, for example, that there is no reasonable structure theorem for $\aleph_1$-saturated models of this theory.
The order sequence of a finite group $G$ is a non-decreasing finite sequence formed of the element orders of $G$. Several properties of order sequences were studied by P. J. Cameron and H. K. Dey in a recent paper that concludes with a list…
Broadly speaking, a finiteness property of groups is any generalisation of the property of having finite order. A large part of infinite group theory is concerned with finiteness properties and the relationships between them. Profinite…
We provide a necessary and sufficient condition for separability of Gaussian states of bipartite systems of arbitrarily many modes. The condition provides an operational criterion since it can be checked by simple computation. Moreover, it…
This paper critically examines arguments against independence, a measure of group fairness also known as statistical parity and as demographic parity. In recent discussions of fairness in computer science, some have maintained that…
We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that free groups cannot be given discrete orders, although they do have right-orders which are…
This paper is the first in a sequence on the first order theory of free products. In the first paper we generalize the analysis of systems of equations over free and (torsion-free) hyperbolic groups, and analyze system of equations over…
In this paper, we study the class of free multiarrangements of hyperplanes. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over the rationals.
Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we…
This is a survey and research note on the modified Orlik conjecture derived from the division theorem introduced in [2]. The division theorem is a generalization of classical addition-deletion theorems for free arrangements. The division…
We are concerned with orderable groups and particularly those with orderings invariant not only under multiplication, but also under a given automorphism or family of automorphisms. Several applications to topology are given: we prove that…
In this paper, we will prove some sufficient conditions for the solvability of groups.
We define what is meant by a strict total order in a category having subobjects, products and fibre products. This allows us to define the notions of an ordered bundle X and an ordered G-set; when G=\pi_1(X) we relate these structures to…
In this paper, we will construct the graph free product of noncommutative probability space. This is the attempt to explain and observe the combinatorial-object-depending probabilistic structure.
In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.