Menger algebras of terms induced by transformations with restricted range

Abstract

In this paper, a special kind of n-ary terms of type τn, which are called T (¯n,Y)-full terms, are introduced. They are derived by applying transformations on the set ¯n = {1, 2, …, n} with restricted range. Under the superposition operation Sn, the algebra of such terms called the clone of T (¯n,Y)-full terms is constructed. We prove that the superassociative law is satisfied in the clone of T (¯n,Y)-full terms and the freeness is investigated using a generating set and a suitable homomorphism. Based on the theory of hypervariety, we study T (¯n,Y)-full hypersubstitutions which are maps taking all operation symbols to our obtained terms. These lead us to provide the classes of T (¯n,Y)-full hyperidentities and T (¯n,Y)-full solid varieties. A connection between identities in cloneT(¯n,Y) (τn) and T (¯n,Y)-full hyperidentities is established.