Uniform $m$-equivalences and numberings of classical systems

The paper considers the representability of algebraic structures (groups, lattices, semigroups, etc.) over equivalence relations on natural numbers. The concept of a (uniform) $m$-equivalence is studied. It is proved that the numbering equivalence of any numbered group is a uniform $m$-equivalence. On the other hand, we construct an example of a uniform $m$-equivalence over which no group is representable. Additionally we show that there exists a positive equivalence over which no upper (lower) semilattice is representable.