Finite Semigroups Satisfying an Identity $x_1\dots x_n\approx \rho (x_1,\dots ,x_n)$

Abstract

We determine the maximal pseudovarieties of finite semigroups that satisfy an identity of the form $x_1\dots x_n\approx \rho (x_1,\dots ,x_n)$. Applying this classification, we further show that a pseudovariety of permutative semigroups satisfies a common permutation identity if and only if it satisfies an identity of the above form or, equivalently, if it does not contain $\mathbf{T}=[x^2\approx xyx\approx 0]$.