Clones of full terms

Abstract

In this paper the well-known connection between hyperidentities of an algebra and identities satisfied by the clone of this algebra is studied in a restricted setting, that of \(n\)-ary full hyperidentities and identities of the \(n\)-ary clone of term operations which are induced by full terms. We prove that the \(n\)-ary full terms form an algebraic structure which is called a Menger algebra of rank \(n\). For a variety \(V\), the set \(Id_n^FV\) of all its identities built up by full \(n\)-ary terms forms a congruence relation on that Menger algebra. If \(Id_n^FV\) is closed under all full hypersubstitutions, then the variety \(V\) is called \(n-F-\)solid. We will give a characterization of such varieties and apply the results to \(2-F-\)solid varieties of commutative groupoids.