Partial clones determined by concatenated relations

Let k≥2, k be a k-element set, ρ₁ and ρ₂ two relations on k and let ρ₁⊖ρ₂ be the concatenation of ρ₁ and ρ₂. We study the link between the partial clones pPol ρ₁∩pPol ρ₂ and pPol (ρ₁⊖ρ₂). Using results arising from this study we address the following problem: given two maximal partial clones M₁ and M₂ over k, under what conditions is the partial clone M₁∩M₂ covered by M₁ or by M₁? So far the research in this direction was focused on partial clones of Boolean functions and on Slupecki type maximal partial clones.