Prime (−1,1) and Jordan monsters and superalgebras of vector type

It is proved that the prime degenerate (−1,1) algebra constructed in [12] (the (−1,1)-monster) generates the same variety of algebras as the Grassmann (−1,1)-algebra. Moreover, the same variety is generated by the Grassmann envelope of any simple nonassociative (−1,1)-superalgebra. The variety occurs to be the smallest variety of (−1,1)-algebras that contains prime nonassociative algebras.Similar results are obtained for Jordan algebras. Thus, the Jordan monster (the prime degenerate algebra constructed in [12]) and the Grassmann envelope of the prime Jordan superalgebra of vector type have the same ideals of identities. It is also shown that the Jordan monster generates a minimal variety that contains prime degenerate Jordan algebras.All the algebras and superalgebras are considered over a field of characteristic 0.