1. Recently I. Gelfand1 has defined a class of characters of Abelian groups embedded in a normed commutative vector ring. The present note contains an elaboration of this theory with a strengthening of the main theorem and extensions to semi-groups. Let 9R be a normed commutative vector ring with unit element e, £ the set of maximal ideals 9 in 9?, 9/9) the corresponding residue class rings eac...

The ‘degree of k-step nilpotence’ a finite group G is the proportion tuples (x1,…, xk+1 ∈ Gk+1 for which simple commutator [x1, …, xk+1] equal to identity. In this paper we study versions an infinite G, with degree nilpotence defined by sampling in various natural ways, such as random walk, or Følner sequence if amenable. our first main result show that finitely generated, then positive and onl...

Metric algebras are metric variants of Σ-algebras. They are first introduced in the field of universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. Recently a similar notion of quantitative algebra is used in computer science to formulate computational effects of probabilistic programs. In this paper we show that varieties of metric algebras (class...

A dilatation structure encodes the approximate self-similarity of a metric space. A metric space (X, d) which admits a strong dilatation structure (definition 2.2) has a metric tangent space at any point x ∈ X (theorem 4.1), and any such metric tangent space has an algebraic structure of a conical group (theorem 4.2). Particular examples of conical groups are Carnot groups: these are simply con...

We will present a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topo...

We introduce various probabilistic finiteness conditions for profinite groups related to positive finite generation (PFG). investigate completed group rings which are PFG as modules, and use this answer [17, Question 1.2] on positively finitely groups. Using the theory of projective covers, we define characterise a version FPn property groups, called PFPn. Finally, prove how these previously de...

The aim of the present paper is to prove that the family of all closed nonempty subsets of a complete probabilistic metric space L is complete with respect to the probabilistic Pompeiu-Hausdorff metric H . The same is true for the families of all closed bounded, respectively compact, nonempty subsets of L. If L is a complete random normed space in the sense of Šerstnev, then the family of all n...