On some varieties of ai-semirings satisfying \(x^{p+1} \approx x\)
A semiring is called additively idempotent if its additive reduct is a semilattice. \textit{M. M. Ren} et al., [Algebra Univers. 77, No. 4, 395--408 (2017; Zbl 1371.08005)] have described the lattice \(L_2\) of varieties of additively idempotent semirings satisfying the identity \(x^3\approx x\). It was proved that \(L_2\) is discributive and consists of 179 varieties, each of which is finitely based and finitely generated. If an additively idempotent semiring satisfies \(x^3\approx x\), its multiplicative reduct is a union of groups of exponent 2. The authors fix a prime number \(p\) and consider the lattice \(L_p\) of varieties of additively idempotent semirings whose multiplicative reducts are unions of abelian groups of exponent \(p\). They prove that \(L_p\) is discributive and consists of 179 varieties, each of which is finitely based and finitely generated (Theorems 3.16 and 3.17). Reviewer's remark. Even though the authors do not state this explicitly, their proofs show that the lattice \(L_p\) does not really depend on \(p\): all lattices of this form are isomorphic.
scientific article; zbMATH DE number 7045456
scientific article