Decompositions of group functions with applications to two-rail cascades

Abstract

Consider functions from some finite nonempty set X into some finite semigroup S. Let Xm denote the mth Cartesian power of X. A notion of decomposability of functions f: Xm → S into functions from X into S is defined. Necessary and sufficient conditions for the decomposability of f are derived in the special case where S = S3 or S4, the symmetric groups on three and four objects, respectively, and X = {0, 1}. These results are applied to show that any two-output combinational switching network is realizable by a two-rail cascade.