Hyperidentities of Dyadic Algebras

Hyperidentities of universal algebras are related to identities of clones. Clones of functions defined on a set A can be characterized as sets of functions which are closed with respect to composition and which contain all projections f l : (xl, . . . , xn) --+ xl , i = 1, . . . , n, n E N ([Z]). For a universal algebra A = (A ; F) let T ( A ) be the clone of all term functions generated by the set F of fundamental operations of A and the projections. Clones can be considered as universal algebras of type (2, 1 , 1 , 1 , 0 ) ([7], [14]). Therefore it makes sense to study identities for clones called clone equations in [15] and [12]. A hyperidentity a = e is formally the same as an ordinary identity of formal terms a and e of operation symbols and variables, but we say, an algebra, A = (A; F) (or a class K of algebras) satisfies the hyperidentity CT = e, in sign A k a = e (K C a = e), if a = e holds identically in A (in every algebra of K) for every choice of term functions of A (of every algebra in K) with appropriate arities to represent the operation symbols appearing in a and e. Let 0, be the set of all functions defined on A. A finite nontrivial algebra A = (A; F) is said to be primal if T ( A ) = 0,. The two-element Boolean algebra 2 = ( (0 , l}; A, v, +, N) (where A, v, =>, N denote conjunction, disjunction, implication, and negation) is an important example for a primal algebra. The variety V(2) generated by 2 is the variety of all Boolean algebras. Boolean algebras are examples for dyadic abebras, i.e. algebras which are isomorphic to a subdirect product of two-element algebras ([l]).