부산대학교 도서관

Abstract We show that for each $$n\ge 2\in {\mathbb {N}}$$ n ≥ 2 ∈ N , the varieties $${\mathbb {V}}_{n}=[x_1x_2x_3=x_1^nx_{i_1}x_{i_2}x_{i_3}]$$ V n = [ x 1 x 2 x 3 = x 1 n x i 1 x i 2 x i 3 ] where i is any non-trivial permutation of $$\{1,2,3\}$$ { 1 , 2 , 3 } are closed. Further, we show that for each $$n\in {\mathbb {N}}$$ n ∈ N , the varieties $${\mathcal {V}}_{n}=[x_1x_2x_3=x_1^nx_{i_1}x_{i_2}x_{i_3}]$$ V n = [ x 1 x 2 x 3 = x 1 n x i 1 x i 2 x i 3 ] where i is any non-trivial permutation of $$\{1,2,3\}$$ { 1 , 2 , 3 } other than the permutation (231) are closed.