Related papers: Partial associativity and rough approximate groups

We extend several techniques and theorems from geometric group theory so that they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). A second way that we generalize earlier results is by eliminating freeness…

In this paper we define a class of braces, that we call module braces or $R$-braces, which are braces for which the additive group has also a module structure over a ring $R$, and for which the values of the gamma functions are…

In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie…

We partially answer two questions of Goodaire by showing that in a finite, strongly right alternative ring, the set of units (if the ring is with unity) is a Bol loop under ring multiplication, and the set of quasiregular elements is a Bol…

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map sending (x,y) to the product of the Nth powers of x and y is…

In the context of general rough sets, the act of combining two things to form another is not straightforward. The situation is similar for other theories that concern uncertainty and vagueness. Such acts can be endowed with additional…

In this paper, we study the category of trigroups as a generalization of the notion of digroup [4] and analyze their relationship with 3-racks [1] and Leibniz 3-algebras [6]. Trigroups are essentially associative trioids in which there are…

For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger…

We show that if a finite point set $P\subseteq \mathbb{R}^2$ has the fewest congruence classes of triangles possible, up to a constant $M$, then at least one of the following holds. (1) There is a $\sigma>0$ and a line $l$ which contains…

An associative central simple algebra is a form of matrices, because a maximal \'{e}tale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of…

We show that every irreducible, simply connected curve on a toric affine surface X over the field of complex numbers is an orbit closure of a multiplicative group action on X. It follows that up to the action of the automorphism group…

In this paper we have considered a finite unitary matrix group with exact elements being unknown and only approximate elements available. Such a group becomes inconsistent with its own multiplication table. We found simple correction…

We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is…

The covering number of an associative ring $R$ is the minimal number of proper subrings whose union is $R$. We establish a strategy to classify unital rings of a given finite covering number, and obtain a classification of unital rings…

In this paper, we study half-densities enhancing the multiplication map on a symplectic groupoid and which satisfy a suitable associativity condition. This is structurally motivated by the expected complete semiclassical-analytic…

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every…

Let $G$ be a group with a metric $\mathrm{d}$ invariant under left and right translations, and let $\bar{\mathbb{D}}_r$ be the ball of radius $r$ around the identity. A $(k,r)$-metric approximate subgroup is a symmetric subset $X$ of $G$…

Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case…

An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our…

In this paper, we introduce and develop the circle embedding method. This method hinges essentially on a combinatorial-geometric structure which we choose to call circles of partition. We provide applications in the context of problems that…