On the role of logical connectives for primality and functional completeness of algebras of logics

We consider certain finite universal algebras arising from algebraic semantics in implicational logics. They contain a binary operation image and two constants 0 and 1 satisfying the axioms image and image valid in most implicational logics. We characterize the completeness (also called primality) of such algebras, i.e. the property that every finitary operation on their universe is a term operation of the algebra (in other words, it is a composition of the basic operations of the algebra). Using clone theory and the knowledge of maximal clones we describe completeness (functional completeness) in terms of nonpreservation of three types of specific relations. If image has a simple property and the algebra contains a binary operation image with a neutral element 1 and a unary operation image satisfying image and image otherwise, the algebra is functionally complete.