An extension of a combinatorial fixed point theorem of Shih and Dong

Shih and Dong have proved a boolean analogue of the Jacobian problem: if a map from n{0,1} to itself has the property that all the boolean eigenvalues of the discrete Jacobian matrix of each element of n{0,1} are zero, then it has a unique fixed point. In this note, this result is extended to any map F from the product X of n finite intervals of integers to itself. Our method of proof reveals an interesting property of the asynchronous state graph of F used to model the qualitative behavior of genetic regulatory networks.