and

A clone on a set A is a collection of (finitary) operations on A that contains the projection operations and is closed under composition. If A = 〈A;F1, F2,... 〉 is an algebra of some signature, then the clone of A consists of the term operations of A. An n-ary relation on A is a compatible relation of A if R is a subuniverse of An. The collection of all (finitary) compatible relations of A is called the (finitary) relational clone of A. It is a well known fact that if A is finite, then the relational clone of A determines the clone of A as follows: an operation f belongs to the clone of A if and only if all compatible relations of A are compatible relations of 〈A; f〉. In fact, the clone of A is determined already by the compatible relations that are completely ∩-irreducible and directly indecomposable; we will call these relations critical. Berman, Idziak, Marković, McKenzie, Valeriote, and Willard [2] investigated those finite algebras A which have ‘few compatible relations ’ in the following sense: there is a polynomial p(n) such that the number of n-ary compatible relations of A is bounded above by 2p(n). They proved that A has few compatible relations if and only if for some k ≥ 2 there is a (k+ 1)-ary term E in the language of A that satisfies the identities A | = E x x y y · · · y y x x y y y y y x