The Monoid of Hypersubstitutions of Type (n)

Abstract

A hypersubstitution of type (n) is a map σ which takes the n-ary operation symbol f to an n-ary term σ (f). Any such σ can be inductively extended to a map ς on the set of all terms of type (n), and any two such extensions can be composed in a natural way. Thus, the set Hyp(n) of all hypersubstitutions of type (n) forms a monoid. For n = 2, many properties of this monoid were described by Denecke and Wismath [5]. In this paper, we study the semigroup properties of Hyp(n) for arbitrary n ≥ 2. In particular, we characterize the projection, dual and idempotent hypersubstitutions, and describe the classes of these elements under Green’s relations.