Minimization of the Shadows in the Partial Mappings Semilattice

Let g,h be partial mappings of {1, 2, ..., n} into {0, 1, ..., k} and D(g),D(h) be their domains. We say that h is greater or equal to g iff D(h) ⊆ D(g) and g(x) = h(x) for all x ∈ D(h). The collection of all partial mappings with the order just defined forms the ranked poset, which we denote by F n k . We may assign to each mapping g a vector ã = (a1, ..., an) such that ai ∈ {−1, 0, ..., k} where ai = g(i) for i ∈ D(g) and ai = −1 for i 6∈ D(g). So, up to the end of the paper we will not make a difference between mappings and vectors. For ã = (a1, ..., an) denote