All idempotent hypersubstitutions of type (2,2)

Abstract

A hypersubstitution of type (2,2) is a map σ which takes the binary operation symbols f and g to binary terms σ(f) and σ(g). Any such σ can be inductively extended to a map \(\hat{\sigma}\) on the set of all terms of type (2,2). By using this extension on the set Hyp(2,2) of all hypersubstitutions of type (2,2) a binary operation can be defined. Together with the identity hypersubstitution mapping f to f(x1,x2) and g to g(x1,x2) the set Hyp(2,2) forms a monoid. This monoid is isomorphic to the endomorphism monoid of the clone of all binary terms of type (2,2). We determine all idempotent elements of this monoid. The results can be applied to the equational theory of Universal Algebra.