Commutators for near-rings: Huq \({\neq}\) Smith (Q331020)

The authors find a counter-example showing that the Huq commutator does not coincide with the Smith commutator for near rings. These commutators coincide for groups and for rings. The Huq commutator has been defined by \textit{S. A. Huq} in 1968 for semiabelian varieties of universal algebras in a paper published in [Q. J. Math., Oxf. II. Ser. 19, 363--389 (1968; Zbl 0165.03301)] and Smith commutators are defined in the book written by \textit{J. D. H. Smith} [Mal'cev varieties. Berlin-Heidelberg-New York: Springer-Verlag (1976; Zbl 0344.08002)] in 1976.