  Grundy functions have found many applications in a wide variety of games, in
solving relevant problems in Game Theory. Many authors have been working on
this topic for over many years. Since the existence of a Grundy function on a
digraph implies that it must have a kernel, the problem of deciding if a
digraph has a Grundy function is NP-complete, and how to calculate one is not
clearly answered. In this paper, we introduce the concept: Semi-Grundy
function, which arises naturally from the connection between kernel and
semi-kernel and the connection between kernel and Grundy function. We explore
the relationship of this concept with the Grundy function, proving that for
digraphs with a defining hereditary property is sufficient to get a semi-grundy
function to obtain a Grundy function. Then we prove sufficient and necessary
conditions for some products of digraphs to have a semi-Grundy function. Also,
it is shown a relationship between the size of the semi-Grundy function
obtained for the Cartesian Product and the size of the semi-Grundy functions of
the factors. This size is an upper bound of the chromatic number. We present a
family of digraphs with the following property: for each natural number $n\geq
2$, there is a digraph $R_n$ that has two Grundy functions such that the
difference between their maximum values is equal to n. Then it is important to
have bounds for the Grundy or semi-Grundy functions.
