Boolean Max-Co-Clones

In our ISMVL 2012 paper we introduced the notion of max-co-clone as a set of relations closed under a new type of quantification, max-quantification. This new concept was motivated by its connections to approximation complexity of counting constraint satisfaction problems. In this paper we go beyond scattered examples of max-co-clones and describe all max-co-clones on a 2-elements set (Boolean max-co-clones). It turns out that there are infinitely many Boolean max-co-clones and that all of them are regular co-clones, although it is not true for larger sets. Also there are many usual co-clones that are not closed under max-quantification, and therefore are not max-co-clones.