On approximation of topological groups by finite quasigroups and finite semigroups

Abstract

It is known that any locally compact group that is approximable by finite groups must be unimodular. However, this condition is not sufficient. For example, simple Lie groups are not approximable by finite ones. In this paper we consider the approximation of locally compact groups by more general finite algebraic systems. We prove that a locally compact group is approximable by finite semigroups iff it is approximable by finite groups. Thus, there exist some locally compact groups and even some compact groups that are not approximable by finite semigroups. We prove also that whenever a locally compact group is approximable by finite quasigroups (latin squares) it is unimodular. The converse theorem is also true: any unimodular group is approximable by finite quasigroups and even by finite loops. In this paper we prove this theorem only for discrete groups. For the case of non-discrete groups the proof is rather long and complicated and is given in a separate paper.