Homotopes of (−1, 1)-algebras with two generators

We present an example of a (−1, 1)-algebra that has an isotope which is not an (−1, 1)-algebra. We prove that the defining relation is preserved by the homotopes of 2-generated (−1, 1)-algebras and, moreover, that the variety generated by a free (−1, 1)-algebra of rank 2 is stable under the operation of taking a homotope. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 550–556, April, 1996. I wish to thank Professor S. V. Pchelintsev for his help during the preparation of this paper.