On Centralizers of Finite Lattices and Semilattices
Endre Tóth and Tamás Waldhauser
We study centralizer clones of finite lattices and semilattices. For semi- lattices, 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.
Keywords: Centralizer, clone, primitive positive clone, commuting operations, semilattice, lattice, distributive lattice, Post lattice
2010 Mathematics Subject Classiffication. Primary 08A40; Secondary 06A07, 06A12, 06B05, 06B25, 06D05