Subvarieties of the varieties generated by the superalgebra M1,1(E) or M2(K)

Let K be a field of characteristic zero, and let us consider the matrix algebra M-2(K) endowed with the Z(2)-grading (K e(11) circle plus K e(22)) circle plus (K e(12) circle plus K e(21)), We define two superalgebras, R-p and I-q, where p and q are positive integers. We show that if U is a proper subvariety of the variety generated by the superalgebra M-2(K), then the even-proper part of the T-2-ideal of graded polynomial identities of U asymptotically coincides with the even-proper part of the graded polynomial identities of the variety generated by the superalgebra. R-p circle plus I-q. This description also affords an even-asymptotic description of the proper subvarieties of the variety generated by the superalgebra M-1,M-1(E) as even-asymptotically coinciding with the T-2-ideal of the variety generated by the Grassmarm envelopes G(R-p) and G(I-q). Moreover, the following general fact is established. If two varieties of superalgebras are even-asymptotically equivalent, then they are asymptotically equivalent, and they have the same PI-exponent.