The lattice of ai-semiring varieties satisfying xn≈x and xy≈yx

• Published in: Semigroup forum Vol. 100; no. 2; pp. 542 - 567
• Main Authors: Ren, Miaomiao, Zhao, Xianzhong, Shao, Yong
• Format: Journal Article
• Language: English
• Published: New York Springer US 2020; Springer Nature B.V
• Subjects: Algebra; Intervals; Mathematics; Mathematics and Statistics; Research Article; Ai-semiring; Finitely generated variety; Finitely based variety; Identity; Variety; Lattice
• ISSN: 0037-1912; 1432-2137
• DOI: 10.1007/s00233-020-10092-8

We study the lattice L ( CSr ( n , 1 ) ) of subvarieties of the ai-semiring variety CSr ( n , 1 ) defined by x n ≈ x and x y ≈ y x . We divide L ( CSr ( n , 1 ) ) into five intervals and provide an explicit description of each member of these intervals except [ CSr ( 2 , 1 ) , CSr ( n , 1 ) ] . Based on these results, we show that if n - 1 is square-free, then L ( CSr ( n , 1 ) ) is a distributive lattice of order 2 + 2 r + 1 + 3 r , where r denotes the number of prime divisors of n - 1 . Also, all members of L ( CSr ( n , 1 ) ) are finitely based and finitely generated and so CSr ( n , 1 ) is a Cross variety. Moreover, the axiomatic rank of each member of L ( CSr ( n , 1 ) ) is less than four.