ESSENTIAL OPERATION SYMBOLS IN TERMS

Abstract

Generalizing the concept of an essential variable in a term with respect to an algebra or a variety, we define essential operation symbols in a term with respect to an algebra or with respect to a variety of algebras. Using the concept of a unitary Menger algebra of rank n we define socalled operator terms and prove that essential operation symbols in terms with respect to an algebra correspond to essential variables in operator terms with respect to the Menger algebra of all n-ary term operations of this algebra. After proving some elementary propositions, we extend our definition to essential operation symbols in hypersubstitutions and determine some monoids consisting of hypersubstitutions which contain the same essential operation symbols. Using the isomorphism between the monoid of all endomorphisms of the unitary Menger algebra of all n-ary terms of type τ and the monoid of all hypersubstitutions, we obtain an equivalence between essential variables in operator terms with respect to the Menger algebra n−cloneV and essential operation symbols in terms with respect to the variety V .