Weak bases of Boolean co-clones

• We give weak bases of all Boolean co-clones with a finite base.
• We prove that the relations are minimal with respect to set inclusion.
• The weak bases provide insight into the lattice of strong partial clones.

Universal algebra has proven to be a useful tool in the study of constraint satisfaction problems (CSP) since the complexity, up to logspace reductions, is determined by the clone of the constraint language. But two CSPs corresponding to the same clone may still differ substantially with respect to worst-case time complexity, which makes clones ill-suited when comparing running times of CSP problems. In this article we instead consider an algebra where each clone splits into an interval of strong partial clones such that a strong partial clone corresponds to the CSPs that are solvable within the same O(cn)O(cn) bound. We investigate these intervals and give relational descriptions, weak bases, of the largest elements. They have a highly regular form and are in many cases easily relatable to the smallest members in the intervals, which suggests that the lattice of strong partial clones has a simpler structure than the lattice of partial clones.