Generalized function theory on finite dimensionallinear associative algebras over the fieldo freals has pro-duced many forms o f generalized regular1ty Of particularin -terest are the forms o fregularity known as S-regularity and F-regularity. It is well known that functions regular in one senseneed not be regular in the other. Here we prove a transformationtheorem which allows the formation o f F-regular functions from S-regular functions. Furthermore, a classi flcation theorem isproved that determines the algebras for which this transformationproperty holds.