On some varieties of ai-semirings satisfying x p+1 ≈ x

Abstract

Abstract The aim of this paper is to study the lattice of subvarieties of the ai-semiring variety defined by the additional identities x p + 1 ≈ x and z x y z ≈ ( z x z y z ) p z y x z ( z x z y z ) p , $$\begin{array}{} \displaystyle x^{p+1}\approx x\,\,\mbox{and}\,\,zxyz\approx(zxzyz)^{p}zyxz(zxzyz)^{p}, \end{array} $$ where p is a prime. It is shown that this lattice is a distributive lattice of order 179. Also, each member of this lattice is finitely based and finitely generated.