On centralizers of finite lattices and semilattices

Abstract: We study centralizer clones of finite lattices and semilattices. For semilattices, we give two characterizations of the centralizer and also derive formulas for the number of operations of a given essential arity in the centralizer. We also characterize operations in the centralizer clone of a distributive lattice, and we prove that the essential arity of operations in the centralizer is bounded for every finite (possibly nondistributive) lattice. Using these results, we present a simple derivation for the centralizers of clones of Boolean functions.