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Let F be an algebraically closed field of characteristic zero and let (A 1 , ... , A n) be an n-tuple of simple superalgebras such that n ≥ 3 and there exists 1 < r < n such that A r = M k r (F ⊕ cF) , with c 2 = 1 , whereas A i = M k i , l i (F) for all remaining indices. In this paper, we classify the minimal F-superalgebras such that its graded semisimple part is given by A 1 ⊕ ⋯ ⊕ A n . Moreover, we obtain necessary and sufficient conditions to the supervariety generated by such minimal superalgebra be or not minimal of superexponent dim F (A 1 ⊕ ⋯ ⊕ A n). [ABSTRACT FROM AUTHOR]

Giambruno and Zaicev stated that, over a field of characteristic zero, T-ideals of minimal varieties of a fixed exponent have the factoring property. In the present article, we describe necessary and sufficient conditions for the factorability of T2-ideals of minimal supervarieties of a fixed superexponent. In light of the characterization of minimal supervarieties of a fixed superexponent given by Di Vincenzo, da Silva, and Spinelli, the crucial point is the study of the factorability of T2-ideals of the upper-block triangular matrix algebras equipped with an elementary -grading, where are simple superalgebras. We obtain necessary and sufficient conditions for the isomorphism between two superalgebras. We also show that the concept of -regularity establishes a nice connection between the factorability of the T2-ideal of and the number of isomorphism classes of. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero. We study two minimal superalgebras A and B having the same superexponent but such that T ( A) ⫋ T ( B), thus providing the first example of a minimal superalgebra generating a non minimal supervariety. We compare the structures and codimension sequences of A and B. [ABSTRACT FROM AUTHOR]

Letpbe an odd prime number,Fa field of characteristic zero, and letEbe the unitary Grassmann algebra generated by the infinite-dimensionalF-vector spaceL. We determine the bases of the ℤp-graded identities.Moreover we compute the ℤp-graded codimension and cocharacter sequences for the algebraEendowed with any ℤp-grading such thatLis a homogeneous subspace. [ABSTRACT FROM AUTHOR]