On p-schemes with the same degrees of thin radical and thin residue

Let p and n 1 be a prime number and an integer, respectively. In this paper, first we show that any p-scheme whose thin radical and thin residue are equal is isomorphic to a fission of the wreath product of 2 thin schemes. In addition, we characterize association p-schemes whose thin radical and thin residue each have degree equal to p. We also characterize association p-schemes on p^n points whose thin radical and thin residue each have degree equal to p^{n-1}, and whose basis relations each have valency 1 or p^{n-1}. Moreover, we show that such schemes are Schurian.