Embeddings of quasicells of iterative algebras

We study the relation between the projective and totally restricted extensions of preiterative algebras. We prove that each degree 1 projective extension of a quasicell of the algebra $\mathcal P^*_k$ is a maximal subalgebra of a degree 1 totally restricted extension of the same quasicell. We show also that a projective extension of a quasicell can always be distinguished from its totally restricted extension in the same algebra by hyperidentities.