Cayley-like representations are for all algebras, not merely groups. (Q1771887)

Cayley's Theorem represents a group as a set of permutations with the group operation captured by the composition. Other examples with related representations are monoids, Boolean algebras or Menger algebras, where permutations are replaced by functions with one or more variables. The author shows that any finite algebra has such a representation and so any variety generated by one finite subdirectly irreducible algebra. Such algebras are, e.g., semilattices, distributive lattices, median algebras or elementary Abelian \(p\)-groups. The most general result is obtained for a variety \(\mathcal V\) with a finite number of SI-members which all are finite.