Some large classes of n-valued Sheffer functions

Abstract

In 1967, Singer (11) gave 3 classes of n-valued two-place functors and proved that all these functors were Sheffer functions. Out of the n possible assignments needed to define a functor completely, Singer showed that it was sufficient to define 3n − 2, 3n − 2, and 2n assignments respectivelyfor the 3 classes. We shall enlarge Singer's classes to give functors of type Ia, type II and type III. For types Ia and III, it will be shown that it is sufficient to define 2n − 1 assignments and for type II we require 2n − 1 assignments to be defined and conditions on a further n/p1 assignments (where P1 is the least prime factor of n). These classes of functors include all of Singer's classes. We also introduce functors of type Ib, similar to those of type Ia, and show that for these itis sufficient to define 2n − 1 assignments to ensure the functor is a Sheffer function.