Weak completeness and Abelian semigroups

by T. C. WESSELKAMPER Blacksburg, Virginia (U.S.A.) Let k be a natural number ( k 2 3) and let E(k) = (0, 1, . . . , k 1). Let moreover c‘ = {ha I ha: E ( k ) + {a}}, the set of constant functions. Let En(k) = E(k) x x E(k) , the Cartesian product of n copies of E ( k ) . If f : En(k) -+ E(k) , then say that f is an n place function over E(k) . A set, A of functions over E ( k ) is complete if for each natural number n any n place function f can be exprcsscd as a composition of the functions of A . The set, A is weakly complete if the set A u C is complete. I f A = { f } is complete, then f is Shffer. If A = { f } is weakly complete, then f is weakly Sheffer. This last notion differs from the notion if pseudo-Sheffer used by ROSE [l]. For each j (0 5 j I k 1 ) let V j : E ( k ) + E ( k ) be the function: 1, if x = j; 1 0, if x =t= j. v jx =