Essential operations in centralizer clones

Abstract.

For an algebra A and for any finite n ≥ 2, a homomorphism $h:A^{n} \rightarrow A$ is essentially n-ary if it depends on all its variables. The existence of such an h brings the integer n into the set S(cA) of all significant arities of the centralizer clone cA of the algebra A. For idempotent algebras, algebras with zero and congruence distributive algebras, the set S(cA) must be an order ideal in ω = {0, 1, . . . } or in ω \ {0} or in ω \ {0, 1}, and we construct such algebras. On the other hand, there exist algebras with two unary operations whose centralizer clones have essential arities that form an order filter of ω. We also construct (0, 1)-lattices for which S(cA) is a prescribed finite order ideal of ω \ {0} or of ω \ {0, 1}, and whose endomorphism monoid End₀₁ A of all its (0, 1)-endomorphisms is isomorphic to a prescribed one. Finally, for a product K × L of (0, 1)-lattices (and other algebras), we give conditions under which the set S(c(K × L)) is the smallest possible, and show why these conditions are needed. In the process we prove that for any lattice A, the category ${\mathbb{L}}[A]$ of all lattices containing A and of their lattice homomorphisms is almost universal.