On regular 𝐺-gradings

Let A A be an associative algebra over an algebraically closed field F F of characteristic zero and let G G be a finite abelian group. Regev and Seeman introduced the notion of a regular G G -grading on A A , namely a grading A = ⨁ g ∈ G A g A= g∈ GAg that satisfies the following two conditions: ( 1 ) (1) for every integer n ≥ 1 n≥ 1 and every n n -tuple ( g 1 , g 2 , … , g n ) ∈ G n (g₁,g₂, ,gₙ)∈ Gⁿ , there are elements, a i ∈ A g i