Mutations of \({\mathfrak {perm}}\) algebras

We describe mutation elements in free \(\mathfrak {perm}\) $\mathfrak{perm}$ algebras. Moreover, we construct a base of free mutation of free \(\mathfrak {perm}\) $\mathfrak{perm}$ algebra. Using Cohn’s criterion for the speciality of algebras, we show that there is an exceptional homomorphic image of mutation of free \(\mathfrak {perm}\) $\mathfrak{perm}$ algebras.