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We give a new characterization of partial groups as a subcategory of symmetric (simplicial) sets. This subcategory has an explicit reflection, which permits one to compute colimits in the category of partial groups. We also introduce the…
The aim of this paper is to describe the definitions and main properties of three generalizations of the group concept, namely: groupoid, generalized group and almost groupoid. Some constructions of these algebraic structures and…
A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more…
Different group structures which underline the integrable systems are considered. In some cases, the quantization of the integrable system can be provided with substituting groups by their quantum counterparts. However, some other group…
In this paper we discuss generalized group, provides some interesting examples. Further we introduce a generalized module as a module like structure obtained from a generalized group and discuss some of its properties and we also describes…
We present a generalization of the notion of an algebra norm relevant to real finite-dimensional unital associative algebras. Among other things, this leads to a novel set of algebra isomorphism invariants, some of which are computationally…
In answer to a question of P. Hall, we supply another construction of a group which is isomorphic to each of its non-trivial normal subgroups.
In this article, we compare two different notions of partially defined group strutures, namely partial groups and pregroups, as introduced by Chermak and Stallings respectively. In particular we prove that the category of pregroups can be…
The aim of this paper is to present the main constructions of the substructures of an almost groupoid and to discuss their basic properties. The definitions and properties concerning these new algebraic constructions extend to almost…
This paper aims to examine the version of the topological group structure in proximity and especially descriptive proximity spaces, that is, the concepts of proximal group and descriptive proximal group are introduced. In addition, the…
We explore the notion of sectional number of a group homomorphism, leading to a generalization of the covering number of a group, and present several characterizations when the sectional number is finite, providing estimates for computing…
We provide a treatment of isomorphism within a set-theoretic formulation of dependent type theory. Type expressions are assigned their natural set-theoretic compositional meaning. Types are divided into small and large types --- sets and…
It is known that any covering space of a topological group has the natural structure of a topological group. This article discusses a noncommutative generalization of this fact. A noncommutative generalization of the topological group is a…
We define the Born group as the group of transformations that leave invariant the line element of Minkowski's spacetime written in terms of Fermi coordinates of a Born congruence. This group depends on three arbitrary functions of a single…
We survey structures endowed with natural partial orderings and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism order…
We present a variation of quasi-isometry to approach the problem of defining a geometric notion equivalent to commensurability. In short, this variation can be summarized as "quasi-isometry with uniform parameters for a large enough family…
We work out the theory of fractional isomorphism of graphons as a generalization to the classical theory of fractional isomorphism of finite graphs. The generalization is given in terms of homomorphism densities of finite trees and it is…
Structures of commuting semigroups of isometries under certain additional assumptions like double commutativity or dual double commutativity are found.
Isomorphism is central to the structure of mathematics and has been formalized in various ways within dependent type theory. All previous treatments have done this by replacing quantification over sets with quantification over groupoids of…
We generalize several comparison results between algebraic, semi-topological and topological K-theories to the equivariant case with respect to a finite group.