A class of Sheffer functions obtained by \(\Delta\)-composition of functions (Q1100487)

For \(n\geq 2\), a function \(f:E(n)^ k\to E(n)=\{1,...,n\}\) is called a Sheffer function if any function of E(n) can be obtained by the composition of f. The \(\Delta\)-composition of a binary function \(\circ\) and a unary function T on the set \(E(n)=\{1,...,n\}\) is the binary function \(\phi\) defined by \({\bar \phi}(x,y)=(1-x(x,y))\circ (x,y)+x(x,y)T(x)\) where x is the characteristic function of the diagonal set \(\Delta\) of \(E(n)^ 2\). We show that if T is a cyclic permutation and \(<E(n);\circ >\) either (i) possesses a zero (ii) possesses an identity, or \((ii')\) possesses a left (or right) identity and is right (or left) cancellative, then the \(\Delta\)-composition of \(\circ\) and T is a Sheffer function. An example is provided to show that the \(\Delta\)- composition of a cyclic permutation and a left cancellative with left identity function need not be a Sheffer function.