Orders of Discriminator Classes in Multivalued Logic

For $k\ge2$, discriminator classes, that is, closed classes of functions of $k$-valued logic containing the ternary discriminator $p$, are considered. It is proved that any discriminator class has order at most $\max(3,k)$; moreover, the order of any discriminator class containing all homogeneous functions does not exceed $\max(3,k-1)$, and the order of a discriminator class containing all even functions does not exceed $\max(3,k-2)$. All of these three bounds are attainable.