Related papers: Limits and colimits, generators and relations of p…

There are many examples of `binary' partial groups in the literature: sets equipped an identity and a partially-defined binary operation, such that each element admits an inverse. We show that many of these may be regarded as partial groups…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (a)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every…

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…

Internal categories feature notions of limit and completeness, as originally proposed in the context of the effective topos. This paper sets out the theory of internal completeness in a general context, spelling out the details of the…

We obtain a lifting property for finite quotients of algebraic groups, and applications to the structure of these groups.

We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $\R^n$ in a way that is completely algebraic.…

We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for…

The structure of groups for which certain sets of commutator subgroups are finite is investigated, with a particular focus on the relationship between these groups and those with finite derived subgroup.

We firstly introduce some key concepts in category theory, such as quotient category, completion of limits, $\mathrm{Mor}$ category, and so on; then give the concept of topology algebras and sheaves, and discuss how to restore the structue…

Partial groups are a natural generalization of discrete groups recently introduced by Chermak in connection with the theory of fusion systems. In this paper we develop an extension theory for partial groups based in the classical theory of…

We describe the fully residually F; or limit groups relative to F; (where F is a free group) that arise from systems of equations in two variables over F that have coefficients in F.

An algebraically exact category in one that admits all of the limits and colimits which every variety of algebras possesses and every forgetful functor between varieties preserves, and which verifies the same interactions between these…

Restriction categories were introduced as a way of generalising the notion of partial map categories. In this paper, we define cocomplete restriction category, and give the free cocompletion of a small restriction category as a suitably…

We generalize the idea of cofinite groups, due to B. Hartley. First we define cofinite spaces in general. Then, as a special situation, we study cofinite graphs and their uniform completions. The idea of constructing a cofinite graph starts…

We analyze the recent examples of quantum semigroups defined by M.M. Sadr who also brought up several open problems concerning these objects. These are defined as quantum families of maps from finite sets to a fixed compact quantum…

This paper concerns partial groups, objective partial groups, and (finite) localities, with special attention given to the quotient of a locality by a partial normal subgroup.

We give a topological framework for the study of Sela's limit groups: limit groups are limits of free groups in a compact space of marked groups. Many results get a natural interpretation in this setting. The class of limit groups is known…

The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this paper a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are…

We will explore the nature of when certain finite groups have an equal covering, and when finite groups do not. Not to be confused with the concept of a cover group, a covering of a group is a collection of proper subgroups whose…