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We introduce and discuss a definition of approximation of a topological algebraic system A by finite algebraic systems of some class K. For the case of a discrete algebraic system this definition is equivalent to the well-known definition of a local embedding of an algebraic system A in a class K of algebraic systems. According to this definition A is… (More)
It is known that locally compact groups approximable by finite ones are unimodular, but this condition is not sufficient, for example, the simple Lie groups are not approximable by finite ones as topological groups. In this paper the approximations of locally compact groups by more general finite algebraic systems are investigated. It is proved that the… (More)
In , a class of groups locally embeddable into the class of finite groups (LEF-groups) was introduced and thoroughly investigated. This class arises in a natural way in various approximation problems. First of all, the definition of local embeddability into the class of finite groups is a particular case (for groups with the discrete topology) of the… (More)
Recall that a locally compact group G is called unimodular if the left Haar measure on G is equal to the right one. It is proved in this paper that G is unimodular iff it is approximable by finite quasigroups (Latin squares).
We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem… (More)
We develop an axiomatic set theory — the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to THS, prove consistency of THS and present some applications.
Introduction. The class of groups which is considered in this paper, namely the class of groups loccally embedded into the class of finite groups (LEF-groups), could appear as a specification of one general definition in model theory , but as far as we know, was not considered up to now neither in the group theory, nor in its applications. The first… (More)