Let G be a finite abelian group and let K be an algebraically closed field of characteristic 0. We consider associative unital algebras A over K graded by G, that is \(A=\oplus _{g\in G} A_g\) , where the vector subspaces \(A_g\) satisfy \(A_gA_h\subseteq A_{g+h}\) for every g, \(h\in G\) . Such a G-grading is called regular whenever for every n-tuple \((g_1,\ldots ,g_n)\in G^n\) there exist homogeneous elements \(a_i\in A_{g_i}\) such that \(a_1\cdots a_n\ne 0\) in A; furthermore, for every g, \(h\in G\) and every \(a_g\in A_g\) , \(a_h\in A_h\) one has \(a_ga_h=\beta (g,h)a_ha_g\) for some \(\beta (g,h)\in K^*\) . Here \(\beta (g,h)\) depends only on g and h but not on the elements \(a_g\) and \(a_h\) . It is immediate that \(\beta \) is a skew-symmetric bicharacter on G. The regular decomposition above is minimal if whenever \(\beta (g,h)=\beta (g,k)\) for every \(g\in G\) , then \(h=k\) . In this paper we characterize the generators of the graded variety generated by the natural \(\mathbb {Z}_{2}\) -grading on the Grassmann algebra in terms of \(\mathbb {Z}_{2}\) -graded regular algebras with minimal regular decomposition. Furthermore we describe the finitely generated graded subalgebras of a \(\mathbb {Z}_2\) -graded regular algebra having a minimal regular decomposition. We recall that regular gradings and the corresponding decompositions play an important role in the description of numerical invariants of PI algebras as proved in the papers [1, 4, 6, 26].
