On Completing Partial Groupoids to Semigroups

Let $\mathcal{C}$ be a class of semigroups containing all finite commutative bands, and let p(x) be a real polynomial. The $(\mathcal{C},\mathbf{p})$-completion problem asks whether for a given partial groupoid G there exists a semigroup $S\in\mathcal{C}$ such that G ⊆ S, every product for (a, b) ∈ G2 defined in G coincides with that for (a, b) in S, and |S| ≤ p(|G|). We prove the problem to be ℕℙ-hard in general and ℕℙ-complete if the membership problem for $\mathcal{C}$ is in ℙ.