On the rank of p-schemes

چکیده مقاله

Let n > 1 be an integer and p be a prime number. Denote by ℑ_pn the class of non-thin association p-schemes of degree pⁿ. A sharp upper and lower bounds on the rank of schemes in ℑ_pn with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in ℑ_pn is unique up to isomorphism, and it is a fusion of any association p-schemes with degree pⁿ.