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

Abstract

We consider certain finite universal algebras arising from algebraic semantics in implicational logics. They contain a binary operation -> and two constants 0 and 1 satisfying the axioms 0->x=x->1=x->x=1 and 1->x=x 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 -> has a simple property and the algebra contains a binary operation @? with a neutral element 1 and a unary operation @? satisfying @?(1)=0 and @?x=1 otherwise, the algebra is functionally complete.