The Inclusion Structure of Boolean Weak Bases

Strong partial clones are composition closed sets of partial operations containing all partial projections, characterizable as partial polymorphisms of sets of relations $\Gamma(\mathrm{pPol}(\Gamma))$. If $\mathcal{C}$ is a clone it is known that the set of all strong partial clones whose total component equals $\mathcal{C}$, has a greatest element $\mathrm{pPo}1(\Gamma_{w})$, where $\Gamma_{w}$ is called a weak base. Weak bases have seen applications in computer science due to their usefulness for proving complexity classifications for constraint satisfaction related problems. In this paper we completely describe the inclusion structure between $\mathrm{pPol}(\Gamma_{w}), \mathrm{pPol}(\Delta_{w})$ for all Boolean weak bases $\Gamma_{w}$ and $\Delta_{w}$.