On approximation of topological groups by finite algebraic systems

TL;DR

This paper explores how various classes of locally compact groups can be approximated by finite algebraic systems, revealing limitations and conditions for such approximations, especially regarding finite semigroups and quasigroups.

Contribution

It establishes the equivalence between approximation by finite semigroups and finite groups, and shows that approximation by finite quasigroups implies unimodularity.

Findings

Locally compact groups are approximable by finite left or right quasigroups.

Approximation by finite semigroups is equivalent to approximation by finite groups.

Discrete groups are approximable by finite quasigroups.

Abstract

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 approximation of locally compact groups by finite semigroups is equivalent to approximation by finite groups and thus not all locally compact groups are approximable by finite semigroups. We prove that any locally compact group is approximable by finite left (right) quasigroups but the approximabilty of a locally compact group by finite quasigroups (latin squares) implies its unimodularity. The question if the unimodularity of a locally compact group implies its approximability by finite quasigroups is open. We prove only that the discrete groups are approximable by finite quasigroups.